Concrete mathematics : a foundation for computer science

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352 GENERATING FUNCTIONS Now let’s look again at a sum that has been popping up frequently in this book, S,(n) = Om + 1 m + 2”’ +. . . + (n - 1)” = x km. O
7.6 EXPONENTIAL GENERATING FUNCTIONS 353 And the latter sumI is a geometric progression, so there’s a closed form S(z,n) = $+. (7.77) All we need to do is figure out the coefficients of this relatively simple function, and we’ll know S,i:n), because S,(n) = m! [z”‘]S(z,n). Here’s where 13ernoulli numbers come into the picture. We observed a moment ago that t.he egf for Bernoulli numbers is hence we can write enz-1 S(z) = B(z) - - z = Bo~.+B,~+Bz~+...)(n~+n2~+n3~+-..) ( The sum S,(n) is m! times the coefficient of z”’ in this product. For example, So(n) = O! (h3&) nL n S(n) = 1! Elom+Blm ( > f%(n) = .2! Bo$ + B1 & + B2 &) = ( . . . . * . n; = .!n2-d-n; in3-tn2+in. We have therefore derived the formula 0, = Sz(n) = $n(n - i)(n - 1) for the umpteenth time, and this was the simplest derivation of all: In a few lines we have found the general behavior of S,(n) for all m. The general fo:rmula can be written %-l(n) = &EL,(n) - B,(O)) , where B,(x) is the Bernoulli polynomial defined by B,(x) = t (;)BkX-‘. k (7.78) (7.79) Here’s why: The Bernoulli polynomial is the binomial convolution of the sequence (Bo, B1, B;r, . . . ) with (1, x,x2,. . . ); hence the exponential generating
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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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Page 320 and 321: 7 Generating Functions THE MOST POW
Page 322 and 323: 308 GENERATING FUNCTIONS are added
Page 324 and 325: 310 GENERATING FUNCTIONS (The last
Page 326 and 327: 312 GENERATING FUNCTIONS Now we hav
Page 328 and 329: 314 GENERATING FUNCTIONS The first
Page 330 and 331: 316 GENERATING FUNCTIONS determine
Page 332 and 333: 318 GENERATING FUNCTIONS operation
Page 334 and 335: 320 GENERATING FUNCTIONS Table 320
Page 336 and 337: 322 GENERATING FUNCTIONS middle of
Page 338 and 339: 324 GENERATING FUNCTIONS Fortunatel
Page 340 and 341: 326 GENERATING FUNCTIONS Once we’
Page 342 and 343: 328 GENERATING FUNCTIONS Step 1 is
Page 344 and 345: 330 GENERATING FUNCTIONS nice prope
Page 346 and 347: 332 GENERATING FUNCTIONS Finally, s
Page 348 and 349: 334 GENERATING FUNCTIONS Step 3 was
Page 350 and 351: 336 GENERATING FUNCTIONS This is a
Page 352 and 353: 338 GENERATING FUNCTIONS Table 337
Page 354 and 355: 340 GENERATING FUNCTIONS F(z)' = i
Page 356 and 357: 342 GENERATING FUNCTIONS We can app
Page 358 and 359: 344 GENERATING FUNCTIONS Step 3 is
Page 360 and 361: 346 GENERATING FUNCTIONS Raney’s
Page 362 and 363: 348 GENERATING FUNCTIONS Notice tha
Page 364 and 365: 350 GENERATING FUNCTIONS Chapter 5
Page 368 and 369: 354 GENERATING FUNCTIONS function f
Page 370 and 371: 356 GENERATING FUNCTIONS 7.7 DIRICH
Page 372 and 373: 358 GENERATING FUNCTIONS 5 Find a g
Page 374 and 375: 360 GENERATING FUNCTIONS 20 A power
Page 376 and 377: 362 GENERATING FUNCTIONS 29 What is
Page 378 and 379: 364 GENERATING FUNCTIONS 43 The New
Page 380 and 381: 366 GENERATING FUNCTIONS 53 The seq
Page 382 and 383: 368 DISCRETE PROBABILITY total of 6
Page 384 and 385: 370 DISCRETE PROBABILITY about R if
Page 386 and 387: 372 DISCRETE PROBABILITY The mean o
Page 388 and 389: 374 DISCRETE PROBABILITY more sprea
Page 390 and 391: 376 DISCRETE PROBABILITY hence VS =
Page 392 and 393: 378 DISCRETE PROBABILITY And we can
Page 394 and 395: 380 DISCRETE PROBABILITY variance b
Page 396 and 397: 382 DISCRETE PROBABILITY (1 +z+...
Page 398 and 399: 384 DISCRETE PROBABILITY ~1 and ~2
Page 400 and 401: 386 DISCRETE PROBABILITY where Ug i
Page 402 and 403: 388 DISCRETE PROBABILITY Repeating
Page 404 and 405: 390 DISCRETE PROBABILITY There’s
Page 406 and 407: 392 DISCRETE PROBABILITY But the co
Page 408 and 409: 394 DISCRETE PROBABILITY In the spe
Page 410 and 411: 396 DISCRETE PROBABILITY Here 1 is
Page 412 and 413: 398 DISCRETE PROBABILITY For exampl
Page 414 and 415: 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
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440 ASYMPTOTICS Notice how the 0 te
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442 ASYMPTOTICS Problem 3: The nth
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444 ASYMPTOTICS Problem 4: A sum fr
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446 ASYMPTOTICS Problem 5: An infin
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448 ASYMPTOTICS We might as well do
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450 ASYMPTOTIC3 a rough estimate an
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452 ASYMPTOTICS This last estimate
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454 ASYMPTOTICS Here’s another ex
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456 ASYMPTOTICS On the left is a ty
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458 ASYMPTOTICS (The Bernoulli poly
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460 ASYMPTOTICS The graph of B,(x)
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462 ASYMPTOTICS 9.6 FINAL SUM:MATIO
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464 ASYMPTOTICS The integral JT B4(
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466 ASYMPTOTIC3 where 0 < a,,,, < 1
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468 ASYMPTOTICS In Chapter 5, equat
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470 ASYMPTOTICS For example, we hav
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472 ASYMPTOTICS The final task seem
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474 ASYMPTOTICS This is our approxi
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476 ASYMPTOTIC23 9 Prove (9.22) rig
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478 ASYMPTOTICS 39 Evaluate xOsk
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480 ASYMPTOTICS a What is the asymp
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482 ASYMPTOTIC3 Research problems 6
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484 ANSWERS TO EXERCISES Venn [294]
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486 ANSWERS TO EXERCISES move the b
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488 ANSWERS TO EXERCISES 2.2 This i
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490 ANSWERS TO EXERCISES 2.24 Summi
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492 ANSWERS TO EXERCISES 2.36 (a) B
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494 ANSWERS TO EXERCISES 3.21 If 10
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496 ANSWERS TO EXERCISES the circle
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498 ANSWERS TO EXERCISES 3.40 Let L
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500 ANSWERS TO EXERCISES 3.50 H. S.
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502 ANSWERS TO EXERCISES 4.22 (b,
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504 ANSWERS TO EXERCISES 4.40 Let f
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506 ANSWERS TO EXERCISES 4.50 (a) I
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508 ANSWERS TO EXERCISES and (k- 1)
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510 ANSWERS TO EXERCISES In both ca
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512 ANSWERS TO EXERCISES 5.7 Yes, b
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514 ANSWERS TO EXERCISES because al
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516 ANSWERS TO EXERCISES 5.44 Cance
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518 ANSWERS TO EXERCISES and this i
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520 ANSWERS TO EXERCISES The infini
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522 ANSWERS TO EXERCISES 5.82 Let ~
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524 ANSWERS TO EXERCISES Since ~06j
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526 ANSWERS TO EXERCISJES 5.94 This
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528 ANSWERS TO EXERCISES 6.21 The h
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530 ANSWERS TO EXERCISIES 6.40 If 6
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532 ANSWERS TO EXERCISES solutions
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534 ANSWERS TO EXERCISES 6.62 (a) A
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536 ANSWERS TO EXERCISES 6.74 If p(
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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
Page 598 and 599:
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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