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%I #13 Jul 10 2020 22:10:05
%S 1,119,112681,166923119,302857024681,616967236620839,
%T 1354737230950753441,3135180238488702264959,7543003841027749147438441,
%U 18698821633118804601271092959,47466852090165503045193665276041,122841260732098480578334554450553679,323029586700918689286922557725358306721
%N a(n) = Sum_{k=0..n} (-1)^(n-k) * (n+4*k)!/((n-k)! * k!^5).
%C Diagonal of the rational function 1 / (1 - Sum_{k=1..5} x_k + Product_{k=1..5} x_k).
%F G.f.: Sum_{k>=0} (5*k)!/k!^5 * x^k / (1+x)^(5*k+1).
%t a[n_] := Sum[(-1)^(n - k)*(n + 4*k)!/((n - k)!*k!^5), {k, 0, n}]; Array[a, 13, 0] (* _Amiram Eldar_, Jul 10 2020 *)
%o (PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*(n+4*k)!/((n-k)!*k!^5))}
%o (PARI) N=20; x='x+O('x^N); Vec(sum(k=0, N, (5*k)!/k!^5*x^k/(1+x)^(5*k+1)))
%Y Column k=5 of A336169.
%Y Cf. A082489.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Jul 10 2020
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