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A336214
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a(n) = Sum_{k=0..n} k^n * binomial(n,k)^n, with a(0)=1.
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5
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1, 1, 8, 270, 41984, 30706250, 94770093312, 1336016204844832, 76829717664330940416, 19838680914222199482800274, 20521247958509575370600000000000, 94285013320530947020636486516362047300, 1715947732437668013396578734960052732361179136
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) ~ c * exp(-1/4) * 2^(n^2 - n/2) * n^(n/2) / Pi^(n/2), where c = Sum_{k = -infinity..infinity} exp(-2*k*(k-1)) = exp(1/2) * sqrt(Pi/2) * EllipticTheta(3, -Pi/2, exp(-Pi^2/2)) = 2.036643566277677716389243890291939003151565... if n is even and c = Sum_{k = -infinity..infinity} exp(-2*k^2 + 1/2) = exp(1/2) * EllipticTheta(3, 0, exp(-2)) = 2.096087809957308346119920713317351288828811... if n is odd.
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MATHEMATICA
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Flatten[{1, Table[Sum[k^n*Binomial[n, k]^n, {k, 1, n}], {n, 1, 15}]}]
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PROG
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(PARI) a(n) = if (n==0, 1, sum(k=0, n, k^n * binomial(n, k)^n)); \\ Michel Marcus, Jul 13 2020
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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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