A132092 Numerators of Blandin-Diaz compositional Bernoulli numbers (B^sin)_3,n.

-1, -1, -11, -17, -563, -381, 55277, 242747, 406146379, 104180627, -398489682593, -169622229019, -6523856615663, -251077358513783, 35076901882951197, 2869253069531102351, 20717378005021857058651, 1335883610404565359777223, 27846976637614329871324177

Offset

0,3

Comments

Denominators are A132093. Numerators and denominators given only for even n (odd n have numerators = 0).

References

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

Links

Table of n, a(n) for n=0..18.

Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, see p. 8.

Formula

((x^3)/3!)/(sin(x)-x) = Sum_{n>=0} (B^sin)_3,n ((x^n)/n!).

Maple

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

Mathematica

m = 20;
((x^3)/3!)/(Sin[x]-x) + O[x]^(2m) // CoefficientList[#, x]& // #*Range[0, 2m-2]!& // #[[;; ;; 2]]& // Numerator (* Jean-François Alcover, Mar 23 2020 *)

Prog

(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

Crossrefs

Cf. A132093 (denominators), A132094-A132099.

Sequence in context: A286959 A146446 A228055 * A226258 A056705 A259744

Adjacent sequences: A132089 A132090 A132091 * A132093 A132094 A132095

Keyword

frac,sign

Author

Jonathan Vos Post, Aug 09 2007

Extensions

More terms from R. J. Mathar, May 25 2008

Offset corrected by Andrew Howroyd, Sep 22 2024

Status

approved