A132093 Denominators of Blandin-Diaz compositional Bernoulli numbers (B^sin)_3,n.

1, 10, 350, 1050, 57750, 250250, 2388750, 2231250, 1088106250, 137156250, 105761906250, 2289218750, 8842968750, 51289218750, 45049030468750, 3563716406250, 1099667378906250, 4714260332031250, 14142780996093750

Offset: 0

Author

Jonathan Vos Post, Aug 09 2007

Keywords

• frac
• nonn

Comments

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

Examples

			-1, 0, -1/10, 0, -11/350, 0, -17,1050, 0, -563/57750, 0, -381/250250, 0.
		

References

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

Links

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

Crossrefs

Numerators are A132092.

Cf. A132094-A132099.

Programs

Mathematica

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

PARI

my(N=40, x='x+O('x^N), v=apply(denominator, 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 as suggested by Andrew Howroyd. - N. J. A. Sloane, Sep 22 2024