A132095 Denominators of expansion of e.g.f. x^2/(2*(cos(x)-1)), even powers only.

1, 6, 10, 42, 30, 22, 2730, 6, 34, 798, 330, 46, 2730, 6, 290, 14322, 510, 2, 54834, 6, 4510, 1806, 690, 94, 46410, 66, 530, 798, 174, 118, 56786730, 6, 170, 64722, 30, 1562, 140100870, 6, 2, 474, 230010, 166, 3404310, 6, 20470, 272118, 1410, 2, 900354, 6

Offset: 1

Author

Jonathan Vos Post, Aug 09 2007

Keywords

• frac
• nonn

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

• Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers , Afr. Diaspora J. Math., Volume 7, Number 2 (2008).
• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.

Links

• Robert Israel, Table of n, a(n) for n = 1..1000
• Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, p. 8, 2nd table.

Crossrefs

Numerators are A132094.

Cf. A132092-A132099.

Programs

Maple

A132095 := proc(n) add( 2*(-1)^i*x^(2*i)/(2*i+2)!,i=0..n/2+1) ; denom(coeftayl(-1/%,x=0,n)*n!) ; end: for n from 0 to 46 by 2 do printf("%d, ",A132095(n)) ; od: # R. J. Mathar, Oct 18 2007

Mathematica

A132095[n_] := (s = Sum[ 2*(-1)^i*x^(2*i)/(2*i + 2)!, {i, 0, n/2 + 1}] ; Denominator[SeriesCoefficient[-1/s, {x, 0, n}]*n!]) ;
Table[ A132095[n], {n, 0, 100, 2}] (* Jean-François Alcover, Nov 24 2017, after R. J. Mathar *)

PARI

my(x='x+O('x^100), v=apply(denominator, 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* A132094(n)/(a(n)*x^(2*n-1)) as x -> infinity. - Robert Israel, May 27 2015

Extensions

More terms from R. J. Mathar, Oct 18 2007 and Oct 20 2009

Meaningful name from Joerg Arndt, Jan 25 2024