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%I #22 Jul 13 2020 04:53:13
%S 1,23,2401,347279,58370761,10693893503,2071837562929,417449585719343,
%T 86587926575712937,18366152017597820303,3965385492963153556441,
%U 868598410928920193676023,192552082030654661729957401,43117650276328970463683450639,9738695910884616220689842598481
%N a(n) = Sum_{k=0..n} (-1)^(n-k) * (n+3*k)!/((n-k)! * k!^4).
%C Diagonal of the rational function 1 / (1 - Sum_{k=1..4} x_k + Product_{k=1..4} x_k).
%H Seiichi Manyama, <a href="/A336170/b336170.txt">Table of n, a(n) for n = 0..418</a>
%F G.f.: Sum_{k>=0} (4*k)!/k!^4 * x^k / (1+x)^(4*k+1).
%t a[n_] := Sum[(-1)^(n - k)*(n + 3*k)!/((n - k)!*k!^4), {k, 0, n}]; Array[a, 15, 0] (* _Amiram Eldar_, Jul 10 2020 *)
%o (PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*(n+3*k)!/((n-k)!*k!^4))}
%o (PARI) N=20; x='x+O('x^N); Vec(sum(k=0, N, (4*k)!/k!^4*x^k/(1+x)^(4*k+1)))
%Y Column k=4 of A336169.
%Y Cf. A082488.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Jul 10 2020
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