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A361700
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Constant term in the expansion of (1 + x^4 + y^4 + 1/(x*y))^n.
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2
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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
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OFFSET
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0,7
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LINKS
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FORMULA
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a(n) = Sum_{k=0..floor(n/6)} binomial(2*k,k) * binomial(6*k,2*k) * binomial(n,6*k).
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)
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n) = sum(k=0, n\6, binomial(2*k, k)*binomial(6*k, 2*k)*binomial(n, 6*k));
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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