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%I #26 Jan 24 2024 17:35:59
%S -1,-1,-11,-17,-563,-381,55277,242747,406146379,104180627,
%T -398489682593,-169622229019,-6523856615663,-251077358513783,
%U 35076901882951197,2869253069531102351,20717378005021857058651,1335883610404565359777223,27846976637614329871324177
%N Numerators of Blandin-Diaz compositional Bernoulli numbers (B^sin)_3,n.
%C Denominators are A132093. Numerators and denominators given only for even n (odd n have numerators = 0).
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.
%H Hector Blandin and Rafael Diaz, <a href="https://arxiv.org/abs/0708.0809">Compositional Bernoulli numbers</a>, arXiv:0708.0809 [math.CO], 2007-2008, see p. 8.
%F (((x^3)/3!)/(sin(x)-x) = Sum_{n>=0} (B^sin)_3,n ((x^n)/n!).
%p A132092 := proc(n) local g; g := taylor(sin(x)-x,x=0,n+7) ; g := taylor(g/x^3,x=0,n+4) ; g := taylor( 1/6/g,x=0,n+4) ; n!*coeftayl(g,x=0,n) ; numer(%) ; end: for n from 0 to 40 by 2 do printf("%d,",A132092(n)) ; od: # _R. J. Mathar_, May 25 2008
%t m = 20;
%t ((x^3)/3!)/(Sin[x]-x) + O[x]^(2m) // CoefficientList[#, x]& // #*Range[0, 2m-2]!& // #[[;; ;; 2]]& // Numerator (* _Jean-François Alcover_, Mar 23 2020 *)
%o (PARI) my(N=40, x='x+O('x^N), v=apply(numerator, Vec(serlaplace(x^3/(6*(sin(x)-x)))))); vector(#v\2, k, v[2*k-1]) \\ _Michel Marcus_, Jan 24 2024
%Y Cf. A132093 (denominators), A132094-A132099.
%K frac,sign
%O 1,3
%A _Jonathan Vos Post_, Aug 09 2007
%E More terms from _R. J. Mathar_, May 25 2008
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