A100597 - OEIS

%I #16 Jan 14 2015 16:05:00

%S 1,1,2,5,14,49,258,1385,1342,-13739,1727362,20549165,-892047378,

%T -13084315271,979519187138,16158974238545,-1747908612654946,

%U -32246548780758179,4903305033480792642,100032668564662494485,-20685044415403212103730,-462550882810484735564351

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

%C 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!.

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

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

%H Alois P. Heinz, <a href="/A100597/b100597.txt">Table of n, a(n) for n = 1..100</a>

%e 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.

%p 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

%t 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]!

%t (* _Jean-François Alcover_, Jun 29 2011 *)

%Y Cf. A027641/A027642, A130620, A141411.

%K sign

%O 1,3

%A _Paul Curtz_, Jun 06 2007

%E Edited and more terms from _Alois P. Heinz_, Oct 12 2009