A133002 Numerators of Blandin-Diaz compositional Bernoulli numbers (B^S)_1,n.

1, -1, 5, -1, 139, -1, 859, 71, -9769, 233, -6395527, 145069, -304991568097, -95164619917, 119780081383, -3046785293, 4002469707564917, -102407337854027, 1286572077762833639, 219276930957009857, -20109624681057406222913, 1651690537394493957719

Offset

0,3

Comments

Denominators are A133003. "Bernoulli numbers for S are shown in the table."

The signs of a(0) and a(3) are wrong in table of p. 11 of Bandin article. - Daniel Suteu, Feb 24 2018

Links

Daniel Suteu, Table of n, a(n) for n = 0..200

Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, p. 11, 1st table.

Formula

a(n) = numerator(f(n) * n!), where f(0) = 1, f(n) = -Sum_{k=0..n-1} f(k) / ((n-k+1)!)^2. - Daniel Suteu, Feb 23 2018

E.g.f. for fractions: x / (BesselI(0,2*sqrt(x)) - 1). - Ilya Gutkovskiy, Sep 01 2021

Example

1, -1/4, 5/72, -1/48, 139/21600, -1/540, 859/2540160, 71/483840, -9769/36288000 (corrected by Daniel Suteu, Feb 24 2018).

Mathematica

f[0] = 1; f[n_] := f[n] = -Sum[f[k]/((n-k+1)!)^2, {k, 0, n-1}]; Table[f[n]*n! // Numerator, {n, 0, 21}] (* Jean-François Alcover, Feb 25 2018, after Daniel Suteu *)

Crossrefs

Cf. A132092, A132093, A132094, A132095, A132096, A132097, A132098, A132099.

Cf. A133000, A133001, A133003, A133004, A133005.

Sequence in context: A099740 A362161 A359657 * A272755 A293572 A294257

Adjacent sequences: A132999 A133000 A133001 * A133003 A133004 A133005

Keyword

sign,frac

Author

Jonathan Vos Post, Aug 09 2007

Extensions

Corrected the sign of a(0) and a(3) by Daniel Suteu, Feb 24 2018

Terms beyond a(8) from Daniel Suteu, Feb 24 2018

Status

approved