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A247501 Triangle read by rows, T(n,k) (n>=0, 0<=k<=n) coefficients of the partial fraction decomposition of rational functions generating the columns of A247498 (the Swiss-Knife polynomials evaluated at nonnegative integers).

%I A247501 #9 Jan 16 2015 10:17:17
%S A247501 1,1,1,0,3,2,-2,4,12,6,0,-3,38,60,24,16,-14,60,330,360,120,0,63,2,
%T A247501 1200,3000,2520,720,-272,274,252,3066,17640,29400,20160,5040,0,-1383,
%U A247501 3278,8820,81144,246960,312480,181440,40320
%N A247501 Triangle read by rows, T(n,k) (n>=0, 0<=k<=n) coefficients of the partial fraction decomposition of rational functions generating the columns of A247498 (the Swiss-Knife polynomials evaluated at nonnegative integers).
%F A247501 Let skp_{n}(x) denote the Swiss-Knife polynomials A153641. The T(n,k) are implicitly defined by:
%F A247501 sum_{k=0..n} (-1)^(n+1)*T(n,k)/(x-1)^(k+1) = sum_{k>=0} x^k*skp_n(k).
%F A247501 T(n, 0) = A155585(n).
%F A247501 T(n, n) = A000142(n) = n!.
%F A247501 T(n,n-1)= A001710(n+1) for n>=1.
%e A247501 Triangle starts:
%e A247501 [   1]
%e A247501 [   1,   1]
%e A247501 [   0,   3,   2]
%e A247501 [  -2,   4,  12,    6]
%e A247501 [   0,  -3,  38,   60,    24]
%e A247501 [  16, -14,  60,  330,   360,   120]
%e A247501 [   0,  63,   2, 1200,  3000,  2520,   720]
%e A247501 [-272, 274, 252, 3066, 17640, 29400, 20160, 5040]
%e A247501 .
%e A247501 [n=3] -> [-2,4,12,6] -> -2/(x-1)+4/(x-1)^2+12/(x-1)^3+6/(x-1)^4 = -2*x*(-5*x+x^2+1)/(x-1)^4; g. f. of A247498[n,3] = 0,-2,2,18, ...
%e A247501 [n=4] -> [0,-3,38,60,24] -> 3/(x-1)^2-38/(x-1)^3-60/(x-1)^4-24/(x-1)^5 = (-47*x^2+3*x^3+25*x-5)/(x-1)^5; g. f. of A247498[n,4] = 5,0,-3,32, ...
%p A247501 Trans := proc(T,n) local L, S, k, j, h, r, c;
%p A247501 c := k -> k!*coeff(series(T,t,k+2), t, k);
%p A247501 S := [seq([seq(coeff(c(k),x,j), j=0..k)], k=0..n)];
%p A247501 L := proc(m,k) add(S[m+1][j+1]*k^j, j=0..m) end;
%p A247501 h := sum(x^j*L(n,j), j=0..infinity); r := convert(h, parfrac);
%p A247501 [seq((-1)^(n+1)*coeff(r,(x-1)^(-k-1)), k=0..n)] end:
%p A247501 A247501_row := n -> Trans(exp(x*t)*sech(t), n):
%p A247501 seq(print(A247501_row(n)), n=0..7);
%Y A247501 Cf. A247498, A153641, A155585, A001710.
%K A247501 sign,tabl
%O A247501 0,5
%A A247501 _Peter Luschny_, Dec 14 2014