Concrete mathematics : a foundation for computer science

6.5 BERNOULLI NUMBERS 273
And that’s not all. Bernoulli numbers also appear in the coefficients of
the tangent function,
(6.92)
as well as other trigonometric functions (exercise 70). Formula (6.92) leads
to another important fact about the Bernoulli numbers, namely that
T2n-, = (-1)-l 4n(4n-l)
2n
We have, for example:
n 135 7 9 11 13
Tll 1 2 16 272 7936 353792 22368256
Bzn is a positive integer. (Wi)
(The T's are called tangent numbers.)
One way to prove (6.g3), following an idea of B. F. Logan, is to consider
the power series
sinz+xcosz
- x+ (l+x2)z+ (2x3+2x); + (6x4+8x2+2); +
cosz-xsinz -
When x = tanw, where T,,(x) is a polynomial in x; setting x = 0 gives T, (0) = Tn, the nth
this is tan( z + w) . tangent number. If we differentiate (6.94) with respect to x, we get
1
(cosz-xsinz)2
= xT(x)$;
Tl>O
but if we differentiate with respect to z, we get
1+x2
(cosz-xsin~)~
= tT,(xl& = tT,_M$.
ll>l tl)O
(Try it-the cancellation is very pretty.) Therefore we have
-&,+1(x) = (1 +x2)T;(x), To(x) = x, (fhd
a simple recurrence from which it follows that the coefficients of Tn(x) are
nonnegative integers. Moreover, we can easily prove that Tn(x) has degree
n + 1, and that its coefficients are alternately zero and positive. Therefore
Tz,+I (0) = Tin+, is a positive integer, as claimed in (6.93).