A100597 Based on the first matrix inverse of transformed Bernoulli numbers as defined in the Comments line.

1, 1, 2, 5, 14, 49, 258, 1385, 1342, -13739, 1727362, 20549165, -892047378, -13084315271, 979519187138, 16158974238545, -1747908612654946, -32246548780758179, 4903305033480792642, 100032668564662494485, -20685044415403212103730, -462550882810484735564351

Offset

1,3

Comments

A family of polynomials is defined by P(0,x) = u(0), P(n,x) = u(n) +x*Sum_{i=0..n-1} u(i)*P(n-i-1,x), where u(n) is the n-th Bernoulli number. The coefficients of P(n-1,x) are used to fill the n-th row of the infinite lower triangle matrix M. Then a(n) is given by M^(-1)[n,1] * n!.

References

P. Curtz, Gazette des Mathematiciens, 1992, 52, p.44.

P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, 55, pp.67-78.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..100

Example

a(3) = 2, because M = [1; -1/2 1; 1/6 -1 1; ...], M^(-1) = [1; 1/2 1; 1/3 1 1; ...], and (1/3)*3! = 2.

Maple

P:= proc(n) option remember; local i, u, x; u:= bernoulli; `if`(n=0, u(0), unapply(expand(u(n) +x *add(u(i) *P(n-i-1)(x), i=0..n-1)), x)) end: a:= n-> (1/Matrix(n, (i, j)-> coeff(P(i-1)(x), x, j-1)))[n, 1] *n!: seq(a(n), n=1..30);  # Alois P. Heinz, Oct 12 2009

Mathematica

p[0, x_] = BernoulliB[0]; p[n_, x_] := p[n, x] = BernoulliB[n] + x*Sum[BernoulliB[i]*p[n-i-1, x], {i, 0, n-1}]; t[m_] := Table[ PadRight[CoefficientList[p[n, x], x], m+1], {n, 0, m}]; mmax = 20; Inverse[t[mmax-1]][[All, 1]]*Range[mmax]!
(* Jean-François Alcover, Jun 29 2011 *)

Crossrefs

Cf. A027641/A027642, A130620, A141411.

Sequence in context: A081920 A224978 A006390 * A352184 A022562 A341360

Adjacent sequences: A100594 A100595 A100596 * A100598 A100599 A100600

Keyword

sign

Author

Paul Curtz, Jun 06 2007

Extensions

Edited and more terms from Alois P. Heinz, Oct 12 2009

Status

approved