A132094 Numerators of expansion of e.g.f. x^2/(2*(cos(x)-1)), even powers only.

-1, -1, -1, -5, -7, -15, -7601, -91, -3617, -745739, -3317609, -5981591, -5436374093, -213827575, -213745149261, -249859397004145, -238988952277727, -28354566442037, -26315271553053477373, -108409774812137683, -3394075340453838586663, -62324003400640902910331

Offset: 1

Author

Jonathan Vos Post, Aug 09 2007

Keywords

• frac
• sign

Comments

Numerators and denominators given only for even n (odd n have numerators = 0).

Examples

			-1, 0, -1/6, 0, -1/10, 0, -5/42, 0, -7/30, 0, -15/22, 0, -7601/2730, 0.
		

References

• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.

Links

• Robert Israel, Table of n, a(n) for n = 1..288
• Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, p. 8, 2nd table.

Crossrefs

Denominators are A132095.

Cf. A132092-A132099.

Programs

Maple

A132094 := proc(n) add( 2*(-1)^i*x^(2*i)/(2*i+2)!,i=0..n+1) ; numer(coeftayl(-1/%,x=0,n)*n!) ; end: for n from 0 to 46 by 2 do printf("%d, ",A132094(n)) ; od: # R. J. Mathar, Oct 18 2007

Mathematica

A132094[n_] := (s = Sum[ 2*(-1)^i*x^(2*i)/(2*i + 2)!, {i, 0, n + 1}]; Numerator[SeriesCoefficient[-1/s, {x, 0, n}]*n!]);
Table[A132094[n], {n, 0, 46, 2}] (* Jean-François Alcover, Nov 24 2017, after R. J. Mathar *)

PARI

my(x='x+O('x^50), v=apply(numerator, Vec(serlaplace(x^2/(2*(cos(x)-1)))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Jan 25 2024

Formula

Asymptotic series 2*Psi(1,x) + x*Psi(2,x) ~ Sum_{n>=1} (-1)^n* a(n)/(A132095(n)*x^(2*n-1)) as x -> oo. - Robert Israel, May 27 2015

Extensions

More terms from R. J. Mathar, Oct 18 2007

Meaningful name from Joerg Arndt, Jan 25 2024