A361700 - OEIS

%I #15 Mar 22 2023 07:18:53

%S 1,1,1,1,1,1,31,211,841,2521,6301,13861,30691,90091,360361,1501501,

%T 5645641,18749641,56063281,157520641,445836901,1368402421,4638690211,

%U 16511900791,58059667051,195211574251,625463703151,1942351017751,6016826006101,19113287111101

%N Constant term in the expansion of (1 + x^4 + y^4 + 1/(x*y))^n.

%H Winston de Greef, <a href="/A361700/b361700.txt">Table of n, a(n) for n = 0..1886</a>

%F a(n) = Sum_{k=0..floor(n/6)} binomial(2*k,k) * binomial(6*k,2*k) * binomial(n,6*k).

%F From _Vaclav Kotesovec_, Mar 22 2023: (Start)

%F Recurrence: (n-3)*n^2*(2*n - 9)*(2*n - 3)*a(n) = (24*n^5 - 240*n^4 + 836*n^3 - 1257*n^2 + 843*n - 220)*a(n-1) - (n-1)*(60*n^4 - 600*n^3 + 2094*n^2 - 3051*n + 1600)*a(n-2) + (n-2)*(n-1)*(80*n^3 - 720*n^2 + 2076*n - 1935)*a(n-3) - (n-3)*(n-2)*(n-1)*(60*n^2 - 420*n + 719)*a(n-4) + 24*(n-4)^2*(n-3)*(n-2)*(n-1)*a(n-5) + 725*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6).

%F a(n) ~ sqrt(3/2 + 2^(1/3) + 1/(3*2^(1/3))) * (1 + 3/2^(1/3))^n / (2*Pi*n). (End)

%t Table[Sum[Binomial[2*k,k] * Binomial[6*k,2*k] * Binomial[n,6*k], {k,0,n/6}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 22 2023 *)

%o (PARI) a(n) = sum(k=0, n\6, binomial(2*k, k)*binomial(6*k, 2*k)*binomial(n, 6*k));

%Y Cf. A344560, A361657, A361699.

%K nonn,easy

%O 0,7

%A _Seiichi Manyama_, Mar 21 2023