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

1,2

Comments

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Numerators are A132094.

Cf. A132092-A132099.

Sequence in context: A094890 A213477 A047178 * A332441 A153328 A068588

Adjacent sequences: A132092 A132093 A132094 * A132096 A132097 A132098

Keyword

frac,nonn

Author

Jonathan Vos Post, Aug 09 2007

Extensions

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

Meaningful name from Joerg Arndt, Jan 25 2024

Status

approved