Concrete mathematics : a foundation for computer science

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490 ANSWERS TO EXERCISES 2.24 Summing by parts, t xmHx 6x = ~lnilH,/(m+l)-xm+l/(~+1)2+~; hence &k
A ANSWERS TO EXERCISES 191 There is one solution for each way to write 2100 = x ‘y where x is even and y is odd; we let a == :1x-yI + i and b = i(x+y) + i. So the number of solutions is the number of divisors of 3. 52 .7, namely 12. In general, there are n,,2(n, + 1) ways to represent n,, pn”, where the products range over primes. 2.31 tj,k>2j k = tja21/j2(l -l/j) :=tj,21/j(j-l). Thesecondsumis, similarly, 3/4. 2.32 If2n~x~2~n+l,thesumsareO+~~~+n+(x-n-l)+~~~+(x-2n) = n(x-n) = (x-l) + (x-3) + ... + (x-2n+l). If 2n - 1 < x < 2n they are, similarly, both equa:i to n(x - n). (Looking ahead to Chapter 3, the formula Li(x + l)] (x - [i(x + l)]) covers both cases.) 2.33 If K is empty, AkEK ak = M. The basic analogies are: kEK H /j (C -t ak) = C + A ak kCK kEK ~!ak+bk! = t%+tbk H A min(ak,bk) ktK kcK kEK kEK t kcK jEJ kEK t kEK ak = >, apikl PikICK jEJ kEK ak = ta,[kEK] k &K = min(r\ ak, A bk) kCK H A ak = // aplki PIUEK H A ai.k = A // aj.k iEl ieJ kEK kEK kEK H A ak = Aak.aslkeKi A permutation that 2.34 Let K+ = {k 1ok 3 0} and K = {k 1 ak < 0). Then if, for example, n consumes terms of one sign faster than is odd, we choose F, to be F, -I U E,, where E, g KP is sufficiently large that those of the other 1 kc(F, ,nK* ) ak - t-k@,, tpak) < A can steer the sum toward any value that it l&s. 2.35 Goldbach’s sum can be shown to equal as follows: By unsumming a geometric series, it equals xkEP,La, k ‘; therefore the proof will be complete if we can find a one-to-one correspondence between ordered pairs (m, n) with m,n 3 2 and ordered pairs (k, 1) with k E P and 1 3 1, where m” = k’ when the pairs correspond. If m 4 P we let (m,n) H (m”, 1); but if m = ah E P, we let (m,n) H (an,b). kEK k
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CONCRETE MATHEMATICS Dedicated to L
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CONCRETE MATHEMATICS Ronald L. Grah
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“A odience, level, and treatment
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‘I a concrete life preserver thro
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I’m unaccustomed to this face. De
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n [ n-l 1 n {I m Prestressed concre
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CONTENTS xiii 5.3 Tricks of the Tra
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2 RECURRENT PROBLEMS It’s not imm
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4 RECURRENT PROBLEMS Of course the
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6 RECURRENT PROBLEMS through any of
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8 RECURRENT PROBLEMS From these sma
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10 RECURRENT PROBLEMS But what abou
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12 RECURRENT PROBLEMS that is, in t
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14 RECURRENT PROBLEMS by separating
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16 RECURRENT PROBLEMS For example,
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18 RECURRENT PROBLEMS Homework exer
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20 RECURRENT PROBLEMS 21 Suppose th
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22 SUMS The three-dots notation has
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24 SUMS A slightly modified form of
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26 SUMS where A(n), B(n), and C(n)
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28 SUMS Let’s apply these ideas t
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30 SUMS 2.3 MANIPULATION OF SUMS No
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32 SUMS Typically we use rule (2.20
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34 SUMS because the derivative of a
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36 SUMS instead of summing over all
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38 SUMS The second sum here is zero
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40 SUMS The normal way to evaluate
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42 SUMS First, as usual, we look at
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44 SUMS order to get an equation fo
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46 SUMS One way to use this fact is
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48 SUMS definition where the factor
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50 SUMS Let’s try to recap this;
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52 SUMS we need an appropriate defi
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54 SUMS This formula indicates why
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56 SUMS u(x) = x and Av(x) = 2’;
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58 SUMS The definition in the previ
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60 SUMS In fact, our definition of
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62 SUMS tCj,kiEF a. J, k < A for al
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64 SUMS 18 Let 9%~ and Jz be the re
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66 SUMS 34 35 36 Prove that if the
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68 INTEGER FUNCTIONS below the line
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70 INTEGER FUNCTIONS same inequalit
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72 INTEGER FUNCTIONS An important s
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74 INTEGER FUNCTIONS By the way, th
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76 INTEGER FUNCTIONS (Previously K
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78 INTEGER FUNCTIONS This derivatio
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80 INTEGER FUNCTIONS We’ve got mo
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82 INTEGER FUNCTIONS division, and
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84 INTEGER FUNCTIONS more than one
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86 INTEGER FUNCTIONS 3.5 FLOOR/CEIL
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88 INTEGER FUNCTIONS For this purpo
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90 INTEGER FUNCTIONS it comes out i
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92 INTEGER FUNCTIONS To summarize,
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94 INTEGER FUNCTIONS Also, as we gu
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96 INTEGER FUNCTIONS Basics 10 Show
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98 INTEGER FUNCTIONS 31 Prove or di
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100 INTEGER FUNCTIONS 43 Find an in
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4 Number Theory INTEGERS ARE CENTRA
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104 NUMBER THEORY The left side can
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106 NUMBER THEORY product of primes
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108 NUMBER THEORY only finitely man
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110 NUMBER THEORY But 2P - 1 isn’
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112 NUMBER THEORY arrangements in a
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114 NUMBER THEORY The binary repres
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116 NUMBER THEORY A fraction m/n is
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118 NUMBER THEORY A mediant fractio
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120 NUMBER THEORY This representati
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122 NUMBER THEORY This means that w
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124 NUMBER THEORY Since x mod m dif
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126 NUMBER THEORY than to say that
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128 NUMBER THEORY require division,
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130 NUMBER THEORY Now we must show
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132 NUMBER THEORY And it’s possib
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134 NUMBER THEORY For example, (~(1
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136 NUMBER THEORY Conversely, if f(
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138 NUMBER THEORY (We must add 1 to
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140 NUMBER THEORY The problem of co
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142 NUMBER THEORY n is odd, n4 and
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144 NUMBER THEORY But the inner sum
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146 NUMBER THEORY 22 The number 111
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148 NUMBER THEORY 40 If the radix p
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150 NUMBER THEORY 57 Let S(m,n) be
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152 NUMBER THEORY 73 If the 0(n) +
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154 BINOMIAL COEFFICIENTS For examp
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156 BINOMIAL COEFFICIENTS And now t
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158 BINOMIAL COEFFICIENTS property
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160 BINOMIAL COEFFICIENTS (3, then
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162 BINOMIAL COEFFICIENTS Binomial
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164 BINOMIAL COEFFICIENTS Table 164
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166 BINOMIAL COEFFICIENTS not rows.
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168 BINOMIAL COEFFICIENTS Equation
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170 BINOMIAL COEFFICIENTS that m =
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172 BINOMIAL COEFFICIENTS ifweuse (
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176 BINOMIAL COEFFICIENTS sum on th
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178 BINOMIAL COEFFICIENTS some data
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180 BINOMIAL COEFFICIENTS Problem 4
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182 BINOMIAL COEFFICIENTS We’re l
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184 BINOMIAL COEFFICIENTS We can’
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186 BINOMIAL COEFFICIENTS 5.3 TRICK
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188 BINOMIAL COEFFICIENTS if we app
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190 BINOMIAL COEFFICIENTS The Newto
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192 BINOMIAL COEFFICIENTS One examp
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194 BINOMIAL COEFFICIENTS We can de
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196 BINOMIAL COEFFICIENTS This obse
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198 BINOMIAL COEFFICIENTS Generatin
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200 BINOMIAL COEFFICIENTS can be pu
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204 BINOMIAL COEFFICIENTS This hold
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206 BINOMIAL COEFFICIENTS It’s ou
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208 BINOMIAL COEFFICIENTS into this
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210 BINOMIAL COEFFICIENTS integer.
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212 BINOMIAL COEFFICIENTS into to b
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214 BINOMIAL COEFFICIENTS Now sin(
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216 BINOMIAL COEFFICIENTS But look
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218 BINOMIAL COEFFICIENTS It’s al
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220 BINOMIAL COEFFICIENTS A similar
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222 BINOMIAL COEFFICIENTS One use o
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224 BINOMIAL COEFFICIENTS Therefore
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226 BINOMIAL COEFFICIENTS Now f(-b)
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228 BINOMIAL COEFFICIENTS negative
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230 BINOMIAL COEFFICIENTS valid for
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232 BINOMIAL COEFFICIENTS 18 Find a
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234 BINOMIAL COEFFICIENTS 37 Show t
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236 BINOMIAL COEFFICIENTS 57 Use Go
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238 BINOMIAL COEFFICIENTS 72 Prove
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240 BINOMIAL COEFFICIENTS 86 Let al
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242 BINOMIAL COEFFICIENTS Research
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244 SPECIAL NUMBERS Table 244 Stirl
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246 SPECIAL NUMBERS There are eleve
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248 SPECIAL NUMBERS cycle arrangeme
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250 SPECIAL NUMBERS Table 250 Basic
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252 SPECIAL NUMBERS We can remember
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254 SPECIAL NUMBERS Table 254 Euler
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{'l 'I L\‘, 0 1 / 2 256 SPECIAL N
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258 SPECIAL NUMBERS Table 258 Stirl
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260 SPECIAL NUMBERS ’ ’ ’ (Th
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262 SPECIAL NUMBERS we found that a
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264 SPECIAL NUMBERS Rearranging, we
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266 SPECIAL NUMBERS Thus we have th
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268 SPECIAL NUMBERS (We have used t
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270 SPECIAL NUMBERS Bernoulli numbe
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272 SPECIAL NUMBERS trigonometric f
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274 SPECIAL NUMBERS Recurrence (6.9
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276 SPECIAL NUMBERS Gnethingwecando
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278 SPECIAL NUMBERS cases are: a0 =
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280 SPECIAL NUMBERS The process of
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282 SPECIAL NUMBERS Fk < n < Fk+l;
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284 SPECIAL NUMBERS We have now boi
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286 SPECIAL NUMBERS Before we stop
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288 SPECIAL NUMBERS The continuant
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290 SPECIAL NUMBERS in four steps:
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292 SPECIAL NUMBERS Thus we can con
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294 SPECIAL NUMBERS numbers are 1,
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296 SPECIAL NUMBERS 6 An explorer h
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298 SPECIAL NUMBERS 24 Prove that t
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300 SPECIAL NUMBERS 44 Prove the co
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302 SPECIAL NUMBERS 61 Prove the id
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304 SPECIAL NUMBERS 80 Show that co
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7 Generating Functions THE MOST POW
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308 GENERATING FUNCTIONS are added
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310 GENERATING FUNCTIONS (The last
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312 GENERATING FUNCTIONS Now we hav
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314 GENERATING FUNCTIONS The first
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316 GENERATING FUNCTIONS determine
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318 GENERATING FUNCTIONS operation
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320 GENERATING FUNCTIONS Table 320
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322 GENERATING FUNCTIONS middle of
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324 GENERATING FUNCTIONS Fortunatel
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326 GENERATING FUNCTIONS Once we’
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328 GENERATING FUNCTIONS Step 1 is
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330 GENERATING FUNCTIONS nice prope
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332 GENERATING FUNCTIONS Finally, s
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334 GENERATING FUNCTIONS Step 3 was
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336 GENERATING FUNCTIONS This is a
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338 GENERATING FUNCTIONS Table 337
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340 GENERATING FUNCTIONS F(z)' = i
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342 GENERATING FUNCTIONS We can app
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344 GENERATING FUNCTIONS Step 3 is
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346 GENERATING FUNCTIONS Raney’s
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348 GENERATING FUNCTIONS Notice tha
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350 GENERATING FUNCTIONS Chapter 5
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352 GENERATING FUNCTIONS Now let’
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354 GENERATING FUNCTIONS function f
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356 GENERATING FUNCTIONS 7.7 DIRICH
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358 GENERATING FUNCTIONS 5 Find a g
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360 GENERATING FUNCTIONS 20 A power
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362 GENERATING FUNCTIONS 29 What is
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364 GENERATING FUNCTIONS 43 The New
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366 GENERATING FUNCTIONS 53 The seq
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368 DISCRETE PROBABILITY total of 6
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370 DISCRETE PROBABILITY about R if
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372 DISCRETE PROBABILITY The mean o
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374 DISCRETE PROBABILITY more sprea
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376 DISCRETE PROBABILITY hence VS =
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378 DISCRETE PROBABILITY And we can
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380 DISCRETE PROBABILITY variance b
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382 DISCRETE PROBABILITY (1 +z+...
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384 DISCRETE PROBABILITY ~1 and ~2
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386 DISCRETE PROBABILITY where Ug i
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388 DISCRETE PROBABILITY Repeating
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390 DISCRETE PROBABILITY There’s
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392 DISCRETE PROBABILITY But the co
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394 DISCRETE PROBABILITY In the spe
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396 DISCRETE PROBABILITY Here 1 is
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398 DISCRETE PROBABILITY For exampl
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400 DISCRETE PROBABILITY Thus the m
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402 DISCRETE PROBABILITY Let sj be
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404 DISCRETE PROBABILITY Therefore
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406 DISCRETE PROBABILITY The mean v
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408 DISCRETE PROBABILITY Complicate
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410 DISCRETE PROBABILITY all k and
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412 DISCRETE PROBABILITY to be a pg
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414 DISCRETE PROBABILITY 10 What’
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416 DISCRETE PROBABILITY 28 What is
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418 DISCRETE PROBABILITY 38 What is
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420 DISCRETE PROBABIL1T.Y a Once he
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422 DISCRETE PROBABILITY 50 Conside
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424 DISCRETE PROBABILITY 58 Are the
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426 ASYMPTOTICS The word asymptotic
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428 ASYMPTOTIC3 How about the funct
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430 ASYMPTOTICS N. G. de Bruijn beg
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432 ASYMPTOTICS subtle consideratio
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434 ASYMPTOTICS inefficient “beca
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436 ASYMPTOTICS 9.3 0 MANIPULATION
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138 ASYMPTOTICS Table 438 Asymptoti
Page 454 and 455: 440 ASYMPTOTICS Notice how the 0 te
Page 456 and 457: 442 ASYMPTOTICS Problem 3: The nth
Page 458 and 459: 444 ASYMPTOTICS Problem 4: A sum fr
Page 460 and 461: 446 ASYMPTOTICS Problem 5: An infin
Page 462 and 463: 448 ASYMPTOTICS We might as well do
Page 464 and 465: 450 ASYMPTOTIC3 a rough estimate an
Page 466 and 467: 452 ASYMPTOTICS This last estimate
Page 468 and 469: 454 ASYMPTOTICS Here’s another ex
Page 470 and 471: 456 ASYMPTOTICS On the left is a ty
Page 472 and 473: 458 ASYMPTOTICS (The Bernoulli poly
Page 474 and 475: 460 ASYMPTOTICS The graph of B,(x)
Page 476 and 477: 462 ASYMPTOTICS 9.6 FINAL SUM:MATIO
Page 478 and 479: 464 ASYMPTOTICS The integral JT B4(
Page 480 and 481: 466 ASYMPTOTIC3 where 0 < a,,,, < 1
Page 482 and 483: 468 ASYMPTOTICS In Chapter 5, equat
Page 484 and 485: 470 ASYMPTOTICS For example, we hav
Page 486 and 487: 472 ASYMPTOTICS The final task seem
Page 488 and 489: 474 ASYMPTOTICS This is our approxi
Page 490 and 491: 476 ASYMPTOTIC23 9 Prove (9.22) rig
Page 492 and 493: 478 ASYMPTOTICS 39 Evaluate xOsk
Page 494 and 495: 480 ASYMPTOTICS a What is the asymp
Page 496 and 497: 482 ASYMPTOTIC3 Research problems 6
Page 498 and 499: 484 ANSWERS TO EXERCISES Venn [294]
Page 500 and 501: 486 ANSWERS TO EXERCISES move the b
Page 502 and 503: 488 ANSWERS TO EXERCISES 2.2 This i
Page 506 and 507: 492 ANSWERS TO EXERCISES 2.36 (a) B
Page 508 and 509: 494 ANSWERS TO EXERCISES 3.21 If 10
Page 510 and 511: 496 ANSWERS TO EXERCISES the circle
Page 512 and 513: 498 ANSWERS TO EXERCISES 3.40 Let L
Page 514 and 515: 500 ANSWERS TO EXERCISES 3.50 H. S.
Page 516 and 517: 502 ANSWERS TO EXERCISES 4.22 (b,
Page 518 and 519: 504 ANSWERS TO EXERCISES 4.40 Let f
Page 520 and 521: 506 ANSWERS TO EXERCISES 4.50 (a) I
Page 522 and 523: 508 ANSWERS TO EXERCISES and (k- 1)
Page 524 and 525: 510 ANSWERS TO EXERCISES In both ca
Page 526 and 527: 512 ANSWERS TO EXERCISES 5.7 Yes, b
Page 528 and 529: 514 ANSWERS TO EXERCISES because al
Page 530 and 531: 516 ANSWERS TO EXERCISES 5.44 Cance
Page 532 and 533: 518 ANSWERS TO EXERCISES and this i
Page 534 and 535: 520 ANSWERS TO EXERCISES The infini
Page 536 and 537: 522 ANSWERS TO EXERCISES 5.82 Let ~
Page 538 and 539: 524 ANSWERS TO EXERCISES Since ~06j
Page 540 and 541: 526 ANSWERS TO EXERCISJES 5.94 This
Page 542 and 543: 528 ANSWERS TO EXERCISES 6.21 The h
Page 544 and 545: 530 ANSWERS TO EXERCISIES 6.40 If 6
Page 546 and 547: 532 ANSWERS TO EXERCISES solutions
Page 548 and 549: 534 ANSWERS TO EXERCISES 6.62 (a) A
Page 550 and 551: 536 ANSWERS TO EXERCISES 6.74 If p(
Page 552 and 553: 538 ANSWERS TO EXERCISIES One of th
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540 ANSWERS TO EXERCISES 6.85 The p
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542 ANSWERS TO EXERCISES Hence gzn+
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544 ANSWERS TO EXERCISES 7.24 ntk,+
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546 ANSWERS TO EXERCISES 7.40 The e
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548 ANSWERS TO EXERCISES 1 + a. Hen
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550 ANSWERS TO EXERCISES applied to
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552 ANSWERS TO EXERCISES 8.13 (Solu
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554 ANSWERS TO EXERCISES 8.24 (a) A
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556 ANSWERS TO EXERCISES 8.32 By sy
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558 ANSWERS TO EXERCISES 8.37 The n
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560 ANSWERS TO EXERCISES probabilit
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562 ANSWERS TO EXERCISES And since
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564 ANSWERS TO EXERCISES 8.56 If m
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566 ANSWERS TO EXERCISES We can now
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568 ANSWERS TO EXERCISES 9.19 Hlo =
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570 ANSWERS TO EXERCIS:ES 9.30 Let
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572 ANSWERS TO EXERCISES 9.42 The h
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574 ANSWERS TO EXERCISIES length of
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576 ANSWERS TO EXERCISES 9.58 Let 0
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B Bibliography HERE ARE THE WORKS c
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580 BIBLIOGRAPHY 24' 25 26 27 28 29
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582 BIBLIOGRAPHY 51 52 53 54 55 56
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584 BIBLIOGRAPHY 78 79 80 81 82 83
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586 BIBLIOGRAPHY 100 R. A. Fisher,
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588 BIBLIOGRAPHY 128 Ronald L. Grah
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590 BIBLIOGRAPHY 160 K. Inkeri, “
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592 BIBLIOGRAPHY 188 E.E. Kummer,
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594 BIBLIOGRAPHY 214 Z.A. Melzak, C
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596 BIBLIOGRAPHY 243 George N. Rane
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598 BIBLIOGRAPHY 271 A. D. Solov’
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600 BIBLIOGRAPHY 299 Louis Weisner,
Page 616 and 617:
602 CREDITS FOR EXERCISES 1.1 1.2 1
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604 CREDITS FOR EXERCISES 6.46 6.47
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Index WHEN AN INDEX ENTRY refers to
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608 INDEX Bois-Reymond, Paul David
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610 INDEX Degenerate hypergeometric
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612 INDEX Feder, Tom&s, 604. Feigen
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614 INDEX Hall, Marshall, Jr., 588.
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616 INDEX Lincoln, Abraham, 387. Li
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618 INDEX Patashnik, Oren, iii, iv,
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620 INDEX Running time, 411-412. Ru
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622 INDEX Sylvester, James Joseph,
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List of Tables Sums and differences
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