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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).
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%I #9 Jan 16 2015 10:17:17

%S 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 1200,3000,2520,720,-272,274,252,3066,17640,29400,20160,5040,0,-1383,

%U 3278,8820,81144,246960,312480,181440,40320

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

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

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

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

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

%e Triangle starts:

%e [ 1]

%e [ 1, 1]

%e [ 0, 3, 2]

%e [ -2, 4, 12, 6]

%e [ 0, -3, 38, 60, 24]

%e [ 16, -14, 60, 330, 360, 120]

%e [ 0, 63, 2, 1200, 3000, 2520, 720]

%e [-272, 274, 252, 3066, 17640, 29400, 20160, 5040]

%e .

%e [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 [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 Trans := proc(T,n) local L, S, k, j, h, r, c;

%p c := k -> k!*coeff(series(T,t,k+2), t, k);

%p S := [seq([seq(coeff(c(k),x,j), j=0..k)], k=0..n)];

%p L := proc(m,k) add(S[m+1][j+1]*k^j, j=0..m) end;

%p h := sum(x^j*L(n,j), j=0..infinity); r := convert(h, parfrac);

%p [seq((-1)^(n+1)*coeff(r,(x-1)^(-k-1)), k=0..n)] end:

%p A247501_row := n -> Trans(exp(x*t)*sech(t), n):

%p seq(print(A247501_row(n)), n=0..7);

%Y Cf. A247498, A153641, A155585, A001710.

%K sign,tabl

%O 0,5

%A _Peter Luschny_, Dec 14 2014