A361700 Constant term in the expansion of (1 + x^4 + y^4 + 1/(x*y))^n.
1, 1, 1, 1, 1, 1, 31, 211, 841, 2521, 6301, 13861, 30691, 90091, 360361, 1501501, 5645641, 18749641, 56063281, 157520641, 445836901, 1368402421, 4638690211, 16511900791, 58059667051, 195211574251, 625463703151, 1942351017751, 6016826006101, 19113287111101
Offset: 0
Links
- Winston de Greef, Table of n, a(n) for n = 0..1886
Programs
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Mathematica
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 *) -
PARI
a(n) = sum(k=0, n\6, binomial(2*k, k)*binomial(6*k, 2*k)*binomial(n, 6*k));
Formula
a(n) = Sum_{k=0..floor(n/6)} binomial(2*k,k) * binomial(6*k,2*k) * binomial(n,6*k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
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).
a(n) ~ sqrt(3/2 + 2^(1/3) + 1/(3*2^(1/3))) * (1 + 3/2^(1/3))^n / (2*Pi*n). (End)