Concrete mathematics : a foundation for computer science

6.5 BERNOULLI NUMBERS
6.5 BERNOULLI NUMBERS 269
The next important sequence of numbers on our agenda is named
after Jakob Bernoulli (1654-1705), who discovered curious relationships while
working out the formulas for sums of mth powers [22]. Let’s write
n-1
S,(n) = Om+lm+...+(n-l)m = x km = x;xmsx. (6.77)
(Thus, when m > 0 we have S,(n) = Hi::) in the notation of generalized
harmonic numbers.) Bernoulli looked at the following sequence of formulas
and spotted a pattern:
So(n) = n
S,(n) = 12 ?n - in
Sz(n) = in3 - in2 + in
S3(n) = in4 - in3 + in2
S4(n) = in5 - in4 + in3 - &n
S5(n) = in6 - $5 + fin4 - +pz
k=O
!j6(n) = +n’ - in6 + in5 - in3 + An
ST(n) = in8 - in’ + An6 - &n” + An2
19
&J(n) = Vn
- in8 + $n'- &n5+ $n3- $p
ST(n) = &n’O - in9 + $n8- $n6+ $4- &n2
So(n) = An 11 - +lo+ in9- n7+ n5- 1n3+5n
2 66
Can you see it too? The coefficient of nm+’ in S,(n) is always 1 /(m + 1).
The coefficient of nm is always -l/2. The coefficient of nmP’ is always . . .
let’s see . . . m/12. The coefficient of nmP2 is always zero. The coefficient
of nmP3 is always . . . let’s see . . . hmmm . . . yes, it’s -m(m-l)(m-2)/720.
The coefficient of nmP4 is always zero. And it looks as if the pattern will
continue, with the coefficient of nmPk always being some constant times mk.
That was Bernoulli’s discovery. In modern notation we write the coefficients
in the form
S,(n) = &(Bcnmil + (m:l)B~nm+...+ (m~‘)Bmn)
= &g (mk+‘)BkTlm+l-k.
k=O
(6.78)