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A129334 Triangle T(n,k) read by rows: inverse of the matrix PE = exp(P)/exp(1) given in A011971.

%I A129334 #13 Apr 15 2022 14:57:28
%S A129334 1,-1,1,0,-2,1,1,0,-3,1,1,4,0,-4,1,-2,5,10,0,-5,1,-9,-12,15,20,0,-6,1,
%T A129334 -9,-63,-42,35,35,0,-7,1,50,-72,-252,-112,70,56,0,-8,1,267,450,-324,
%U A129334 -756,-252,126,84,0,-9,1,413,2670,2250,-1080,-1890,-504,210,120,0,-10,1
%N A129334 Triangle T(n,k) read by rows: inverse of the matrix PE = exp(P)/exp(1) given in A011971.
%C A129334 The structure of the triangle is A(r,c) = A000587(1+(r-c))*binomial(r-1,c-1) where row index r and column-index c start at 1.
%C A129334 Row polynomials defined recursively: P(0,x) = 1, P(n+1,x) = x*P(n,x) - P(n,x+1). The polynomials appear to be irreducible. Polynomials evaluated at x = c give sequences with e.g.f. exp(1 - cx - exp(-x)).
%H A129334 S. de Wannemacker, T. Laffey and R. Osburn, <a href="https://arxiv.org/abs/math/0608085">On a conjecture of Wilf</a>, arXiv:math/0608085 [math.NT], 2006-2007.
%F A129334 Let P be the lower-triangular Pascal-matrix, PE = exp(P-I) a matrix-exponential in exact integer arithmetic (or PE = lim exp(P)/exp(1) as limit of the exponential) then A = PE^-1 and a(n) = A(n, read sequentially). - _Gottfried Helms_, Apr 08 2007
%F A129334 T(n, k) = Sum_{j=0..n} (-1)^(j-k)*A094816(j, k)*Stirling2(n, j). - _Mélika Tebni_, Apr 15 2022
%e A129334 Triangle starts:
%e A129334 [0]   1;
%e A129334 [1]  -1,   1;
%e A129334 [2]   0,  -2,    1;
%e A129334 [3]   1,   0,   -3,    1;
%e A129334 [4]   1,   4,    0,   -4,    1;
%e A129334 [5]  -2,   5,   10,    0,   -5,   1;
%e A129334 [6]  -9, -12,   15,   20,    0,  -6,  1;
%e A129334 [7]  -9, -63,  -42,   35,   35,   0, -7,  1;
%e A129334 [8]  50, -72, -252, -112,   70,  56,  0, -8,  1;
%e A129334 [9] 267, 450, -324, -756, -252, 126, 84,  0, -9, 1;
%p A129334 P := proc(n,x) option remember; if n=0 then 1 else
%p A129334 x*P(n-1, x) - P(n-1, x+1) fi end:
%p A129334 aRow := n -> seq(coeff(P(n, x), x, k), k = 0..n):
%p A129334 seq(aRow(n), n = 0..10); # _Peter Luschny_, Apr 15 2022
%Y A129334 First column is A000587 (Uppuluri Carpenter numbers) which is also the negative of the row sums (=P(n, 1)). Polynomials evaluated at 2 are A074051, at -1 A109747.
%Y A129334 Cf. A094816.
%K A129334 easy,tabl,sign
%O A129334 0,5
%A A129334 _Gottfried Helms_, Apr 08 2007
%E A129334 Edited by _Ralf Stephan_, May 12 2007