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A336577
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a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).
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4
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1, 3, 24, 498, 18708, 1055838, 80682414, 7829287392, 924359573112, 128815914107370, 20717986773639696, 3779867347688995698, 771666206195918154156, 174345811623642373266360, 43198501381068549879753648, 11648965476456962547182140512, 3396661425137920919866033312752
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = (1/(n^2+1)) * Sum_{k=0..n} 2^(n-k) * binomial(n^2+1,k) * binomial((n+1)*n-k,n-k).
a(n) ~ 3^n * exp(n + 1/6) * n^(n - 5/2) / sqrt(2*Pi). - Vaclav Kotesovec, Jul 31 2021
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 3^(n-k) * binomial(n,k) * binomial((n+1)*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 3^k * 2^(n-k) * binomial(n,k) * binomial(n^2,k-1) for n > 0. (End)
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MATHEMATICA
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a[n_] := Sum[2^k * Binomial[n, k] * Binomial[n^2 + k + 1, n]/(n^2 + k + 1), {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Jul 27 2020 *)
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PROG
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(PARI) a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(n^2+k+1, n)/(n^2+k+1));
(PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n^2+1, k)*binomial((n+1)*n-k, n-k))/(n^2+1);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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