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A328751 Constant term in the expansion of (-2 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

%I A328751 #14 Mar 20 2023 05:07:40
%S A328751 1,0,30,300,6690,124920,2778600,61790400,1452751650,34806097200,
%T A328751 855836532180,21393889763400,543342862524000,13972938142363200,
%U A328751 363356617578926400,9538720137580233600,252510537115100657250,6733792260826534332000,180751978201192700659500
%N A328751 Constant term in the expansion of (-2 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.
%F A328751 a(n) = Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^5.
%F A328751 From _Vaclav Kotesovec_, Mar 20 2023: (Start)
%F A328751 Recurrence: n^4*(22*n^2 - 198*n + 323)*a(n) = (n-1)*(198*n^5 - 1980*n^4 + 4535*n^3 - 2641*n^2 + 119*n + 210)*a(n-1) + (11066*n^6 - 143858*n^5 + 628715*n^4 - 1298438*n^3 + 1394723*n^2 - 756728*n + 165060)*a(n-2) + 4*(n-2)*(19096*n^5 - 248248*n^4 + 1086158*n^3 - 2156993*n^2 + 2004912*n - 708435)*a(n-3) + 40*(n-3)*(n-2)*(5346*n^4 - 64152*n^3 + 242653*n^2 - 363566*n + 182959)*a(n-4) + 400*(n-4)*(n-3)*(n-2)*(682*n^3 - 6820*n^2 + 17955*n - 12432)*a(n-5) + 6000*(n-5)*(n-4)*(n-3)*(n-2)*(22*n^2 - 154*n + 147)*a(n-6).
%F A328751 a(n) ~ 2^(n-6) * 3^(n+2) * 5^(n + 3/2) / (Pi^2 * n^2). (End)
%t A328751 Table[Sum[(-2)^(n-i)*Binomial[n,i] * Sum[Binomial[i,j]^5, {j,0,i}], {i,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 20 2023 *)
%o A328751 (PARI) {a(n) = sum(i=0, n, (-2)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^5))}
%Y A328751 Column k=5 of A328748.
%Y A328751 Cf. A005261, A328750.
%K A328751 nonn
%O A328751 0,3
%A A328751 _Seiichi Manyama_, Oct 27 2019