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

%I A132095 #35 Jan 25 2024 05:21:32
%S A132095 1,6,10,42,30,22,2730,6,34,798,330,46,2730,6,290,14322,510,2,54834,6,
%T A132095 4510,1806,690,94,46410,66,530,798,174,118,56786730,6,170,64722,30,
%U A132095 1562,140100870,6,2,474,230010,166,3404310,6,20470,272118,1410,2,900354,6
%N A132095 Denominators of expansion of e.g.f. x^2/(2*(cos(x)-1)), even powers only.
%C A132095 Numerators and denominators given only for even n (odd n have numerators = 0).
%D A132095 Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers , Afr. Diaspora J. Math., Volume 7, Number 2 (2008).
%D A132095 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.
%H A132095 Robert Israel, <a href="/A132095/b132095.txt">Table of n, a(n) for n = 1..1000</a>
%H A132095 Hector Blandin and Rafael Diaz, <a href="https://arxiv.org/abs/0708.0809">Compositional Bernoulli numbers</a>, arXiv:0708.0809 [math.CO], 2007-2008, p. 8, 2nd table.
%F A132095 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
%e A132095 -1, 0, -1/6, 0, -1/10, 0, -5/42, 0, -7/30, 0, -15/22, 0, -7601/2730, 0.
%p A132095 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
%t A132095 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!]) ;
%t A132095 Table[ A132095[n], {n, 0, 100, 2}] (* _Jean-François Alcover_, Nov 24 2017, after _R. J. Mathar_ *)
%o A132095 (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
%Y A132095 Numerators are A132094.
%Y A132095 Cf. A132092-A132099.
%K A132095 frac,nonn
%O A132095 1,2
%A A132095 _Jonathan Vos Post_, Aug 09 2007
%E A132095 More terms from _R. J. Mathar_, Oct 18 2007 and Oct 20 2009
%E A132095 Meaningful name from _Joerg Arndt_, Jan 25 2024