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A328809 Constant term in the expansion of (1 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

%I A328809 #11 Oct 28 2019 05:40:07
%S A328809 1,3,39,597,11991,260613,6129489,151078707,3867441111,101852866533,
%T A328809 2744610170049,75348380209347,2100889194001761,59349600029522403,
%U A328809 1695505948476461559,48909452234258070117,1422877722974198091351,41704912707174877940613
%N A328809 Constant term in the expansion of (1 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.
%F A328809 a(n) = Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^5.
%F A328809 From _Vaclav Kotesovec_, Oct 28 2019: (Start)
%F A328809 Recurrence: n^4*(40*n^2 - 24*n - 79)*a(n) = (1080*n^6 - 2808*n^5 + 875*n^4 + 2928*n^3 - 3762*n^2 + 1834*n - 336)*a(n-1) + (9320*n^6 - 42872*n^5 + 61193*n^4 - 12152*n^3 - 35518*n^2 + 21658*n - 2016)*a(n-2) - (n-2)*(48560*n^5 - 223376*n^4 + 216118*n^3 + 381866*n^2 - 791133*n + 355194)*a(n-3) + (n-3)*(n-2)*(79560*n^4 - 286416*n^3 - 56675*n^2 + 976675*n - 616322)*a(n-4) - 11*(n-4)*(n-3)*(n-2)*(5080*n^3 - 8128*n^2 - 25641*n + 21693)*a(n-5) + 363*(n-5)*(n-4)*(n-3)*(n-2)*(40*n^2 + 56*n - 63)*a(n-6).
%F A328809 a(n) ~ 33^(n+2) / (256 * sqrt(5) * Pi^2 * n^2). (End)
%t A328809 Table[Sum[Binomial[n, i]*Sum[Binomial[i, j]^5, {j, 0, i}], {i, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 28 2019 *)
%o A328809 (PARI) {a(n) = sum(i=0, n, binomial(n, i)*sum(j=0, i, binomial(i, j)^5))}
%Y A328809 Column k=5 of A328807.
%Y A328809 Cf. A005261, A328750, A328751.
%K A328809 nonn
%O A328809 0,2
%A A328809 _Seiichi Manyama_, Oct 28 2019