%I A100597
%S A100597 1,1,2,5,14,49,258,1385,1342,13739,1727362,20549165,892047378,
%T A100597 13084315271,979519187138,16158974238545,1747908612654946,
%U A100597 32246548780758179,4903305033480792642,100032668564662494485
%V A100597 1,1,2,5,14,49,258,1385,1342,-13739,1727362,20549165,-892047378,
%W A100597 -13084315271,979519187138,16158974238545,-1747908612654946,
%X A100597 -32246548780758179,4903305033480792642,100032668564662494485
%N A100597 Based on the first matrix inverse of transformed Bernoulli numbers as 
               defined in the Comments line.
%C A100597 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 A100597 P. Curtz, Gazette des Mathematiciens, 1992, 52, p.44.
%D A100597 P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, 
               55, pp.67-78.
%e A100597 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 A100597 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);
%Y A100597 Cf. A027641/A027642, A130620, A141411.
%Y A100597 Sequence in context: A079452 A081920 A006390 this_sequence A022562 A115340 
               A000109
%Y A100597 Adjacent sequences: A100594 A100595 A100596 this_sequence A100598 A100599 
               A100600
%K A100597 sign
%O A100597 1,3
%A A100597 Paul Curtz (bpcrtz(AT)free.fr), Jun 06 2007
%E A100597 Edited with more terms and Maple program by Alois P. Heinz (heinz(AT)hs-heilbronn.de), 
               Oct 12 2009