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A287696 Triangle read by rows, T(n,k) = (n!)^3 * [x^k] [z^n] hypergeom([], [1, 1], z)^x for n>=0, 0<=k<=n.

%I A287696 #19 Jun 13 2017 13:20:42
%S A287696 1,0,1,0,-3,4,0,46,-81,36,0,-1899,3916,-2592,576,0,163476,-375375,
%T A287696 305500,-108000,14400,0,-25333590,63002191,-58725000,26370000,
%U A287696 -5832000,518400,0,6412369860,-16976577828,17470973569,-9168390000,2636298000,-400075200,25401600
%N A287696 Triangle read by rows, T(n,k) = (n!)^3 * [x^k] [z^n] hypergeom([], [1, 1], z)^x for n>=0, 0<=k<=n.
%C A287696 The polynomials Sum_{k=0..n} T(n,k) x^k generate the columns of A287698.
%F A287696 Sum_{k=0..n} T(n,k) = A000012(n).
%F A287696 Sum_{k=0..n} abs(T(n,k)) = A212856(n) = A212855_row(3).
%e A287696 0: [1]
%e A287696 1: [0,         1]
%e A287696 2: [0,        -3,        4]
%e A287696 3: [0,        46,      -81,        36]
%e A287696 4: [0,     -1899,     3916,     -2592,      576]
%e A287696 5: [0,    163476,  -375375,    305500,  -108000,    14400]
%e A287696 6: [0, -25333590, 63002191, -58725000, 26370000, -5832000, 518400]
%p A287696 A287696_row := proc(n) local k; hypergeom([],[1,1],z); series(%^x, z=0, n+1):
%p A287696 n!^3*coeff(%, z, n); seq(coeff(%, x, k), k=0..n) end:
%p A287696 for n from 0 to 8 do A287696_row(n) od;
%p A287696 A287696_poly := proc(n) local k, x; hypergeom([],[1,1],z); series(%^x, z=0, n+1):
%p A287696 unapply(n!^3*coeff(%, z, n), x); end:
%p A287696 for n from 0 to 7 do A287696_poly(n) od;
%t A287696 T[n_, k_] := (n!)^3 SeriesCoefficient[HypergeometricPFQ[{}, {1, 1}, z]^x, {x, 0, k}, {z, 0, n}];
%t A287696 Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 13 2017 *)
%Y A287696 T(n,n) = A001044(n).
%Y A287696 Cf. A212856, A212855, A287314, A287698.
%K A287696 sign,tabl
%O A287696 0,5
%A A287696 _Peter Luschny_, May 30 2017