cp's OEIS Frontend

A355570 Regular triangle of certain polynomial expansion coefficients for the n-th power series.

%I A355570 #15 Aug 24 2022 08:51:14
%S A355570 1,0,1,1,-2,2,0,5,-10,6,1,-10,40,-54,24,0,21,-140,336,-336,120,1,-42,
%T A355570 462,-1764,3024,-2400,720,0,85,-1470,8442,-22176,29520,-19440,5040,1,
%U A355570 -170,4580,-38178,144648,-288000,313200,-176400,40320,0,341,-14080,166452,-875952,2451240,-3920400,3603600,-1774080,362880
%N A355570 Regular triangle of certain polynomial expansion coefficients for the n-th power series.
%C A355570 See the paper by Muschielok for precise definition.
%H A355570 Michel Marcus, <a href="/A355570/b355570.txt">Table of n, a(n) for n = 2..4951</a> (Rows 2..100)
%H A355570 Christoph Muschielok, <a href="https://arxiv.org/abs/2207.01935">Another Approach on Power Sums</a>, arXiv:2207.01935 [math.CO], 2022.
%H A355570 Christoph Muschielok, <a href="https://arxiv.org/abs/2208.11032">Further Identities for cmk- and amk-Weighted Sums and a Remark on a Representation of Pythagoras' Equation</a>, arXiv:2208.11032 [math.NT], 2022.
%e A355570 Triangle begins:
%e A355570   1;
%e A355570   0,   1;
%e A355570   1,  -2,    2;
%e A355570   0,   5,  -10,   6;
%e A355570   1, -10,   40, -54,   24;
%e A355570   0,  21, -140, 336, -336, 120;
%e A355570   ...
%o A355570 (PARI) mat(n) = my(M = matrix(n, n)); M[1, 1] = 1; for (i=2, n, my(p=x + prod(k=-1, i-2, x+k)/(i-2)!); for (j=1, i, M[i, j] = polcoef(p, j, x));); my(iM = 1/M); matrix(n-1, n-1, i, j, iM[i+1, j+1]);
%o A355570 tabl(nn) = {my(m = mat(nn)); for (n=1, nn-1, row = vector(n, k, m[n, k]); print(row, ", "););}
%Y A355570 Cf. A000142 (right diagonal), A202365 (subdiagonal).
%K A355570 sign,tabl
%O A355570 2,5
%A A355570 _Michel Marcus_, Jul 07 2022