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A361699 Constant term in the expansion of (1 + x^3 + y^3 + 1/(x*y))^n.

%I A361699 #16 Mar 22 2023 07:54:51
%S A361699 1,1,1,1,1,21,121,421,1121,2521,6301,23101,99001,386101,1301301,
%T A361699 3943941,11779041,38241841,136988041,504616441,1793870941,6061831441,
%U A361699 19923689941,66139128441,227052188441,800641330721,2831644750221,9870443816221,33869987735221
%N A361699 Constant term in the expansion of (1 + x^3 + y^3 + 1/(x*y))^n.
%H A361699 Winston de Greef, <a href="/A361699/b361699.txt">Table of n, a(n) for n = 0..1798</a>
%F A361699 a(n) = Sum_{k=0..floor(n/5)} binomial(2*k,k) * binomial(5*k,2*k) * binomial(n,5*k).
%F A361699 From _Vaclav Kotesovec_, Mar 22 2023: (Start)
%F A361699 Recurrence: 3*n^2*(3*n - 10)*(3*n - 5)*a(n) = 3*(45*n^4 - 270*n^3 + 510*n^2 - 375*n + 104)*a(n-1) - 45*(n-1)*(6*n^3 - 36*n^2 + 67*n - 40)*a(n-2) + 15*(n-2)*(n-1)*(18*n^2 - 90*n + 109)*a(n-3) - 135*(n-3)^2*(n-2)*(n-1)*a(n-4) + 3152*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
%F A361699 a(n) ~ sqrt(c) * (1 + 5/3^(3/5))^n / (Pi * n), where c = 0.8011502211360696582191471740430432783906089377204901279920664641344364478... is the real root of the equation -2483776 + 28284375*c - 141840000*c^2 + 337500000*c^3 - 405000000*c^4 + 194400000*c^5 = 0. (End)
%t A361699 Table[Sum[Binomial[2*k,k] * Binomial[5*k,2*k] * Binomial[n,5*k], {k,0,n/5}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 22 2023 *)
%o A361699 (PARI) a(n) = sum(k=0, n\5, binomial(2*k, k)*binomial(5*k, 2*k)*binomial(n, 5*k));
%Y A361699 Cf. A344560, A361657, A361700.
%K A361699 nonn,easy
%O A361699 0,6
%A A361699 _Seiichi Manyama_, Mar 21 2023