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

Author

Jonathan Vos Post, Aug 09 2007

Keywords

• frac
• sign

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

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

Crossrefs

Cf. A132093 (denominators), A132094-A132099.

Programs

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 *)

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

Formula

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

Extensions

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

Offset corrected by Andrew Howroyd, Sep 22 2024