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).

1, 1, 1, 0, 3, 2, -2, 4, 12, 6, 0, -3, 38, 60, 24, 16, -14, 60, 330, 360, 120, 0, 63, 2, 1200, 3000, 2520, 720, -272, 274, 252, 3066, 17640, 29400, 20160, 5040, 0, -1383, 3278, 8820, 81144, 246960, 312480, 181440, 40320

Offset: 0

Author

Peter Luschny, Dec 14 2014

Keywords

• sign
• tabl

Examples

			Triangle starts:
[   1]
[   1,   1]
[   0,   3,   2]
[  -2,   4,  12,    6]
[   0,  -3,  38,   60,    24]
[  16, -14,  60,  330,   360,   120]
[   0,  63,   2, 1200,  3000,  2520,   720]
[-272, 274, 252, 3066, 17640, 29400, 20160, 5040]
.
[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, ...
[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, ...
		

Crossrefs

Cf. A247498, A153641, A155585, A001710.

Programs

Maple

Trans := proc(T,n) local L, S, k, j, h, r, c;
c := k -> k!*coeff(series(T,t,k+2), t, k);
S := [seq([seq(coeff(c(k),x,j), j=0..k)], k=0..n)];
L := proc(m,k) add(S[m+1][j+1]*k^j, j=0..m) end;
h := sum(x^j*L(n,j), j=0..infinity); r := convert(h, parfrac);
[seq((-1)^(n+1)*coeff(r,(x-1)^(-k-1)), k=0..n)] end:
A247501_row := n -> Trans(exp(x*t)*sech(t), n):
seq(print(A247501_row(n)), n=0..7);

Formula

Let skp_{n}(x) denote the Swiss-Knife polynomials A153641. The T(n,k) are implicitly defined by:

sum_{k=0..n} (-1)^(n+1)*T(n,k)/(x-1)^(k+1) = sum_{k>=0} x^k*skp_n(k).

T(n, 0) = A155585(n).

T(n, n) = A000142(n) = n!.

T(n,n-1)= A001710(n+1) for n>=1.