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A167010
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a(n) = Sum_{k=0..n} C(n,k)^n.
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24
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1, 2, 6, 56, 1810, 206252, 86874564, 132282417920, 770670360699138, 16425660314368351892, 1367610300690018553312276, 419460465362069257397304825200, 509571049488109525160616367158261124, 2290638298071684282149128235413262383804352
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OFFSET
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0,2
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COMMENTS
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The number of n*n 0-1 matrices with equal numbers of nonzeros in every row. - David Eppstein, Jan 19 2012
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LINKS
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FORMULA
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If n is even then a(n) ~ c * exp(-1/4) * 2^(n^2 + n/2)/((Pi*n)^(n/2)), where c = Sum_{k = -oo..oo} exp(-2*k^2) = 1.271341522189... (see A218792). - Vaclav Kotesovec, Nov 05 2012
If n is odd then c = Sum_{k = -infinity..infinity} exp(-2*(k+1/2)^2) = 1.23528676585389... - Vaclav Kotesovec, Nov 06 2012
a(n) = (n!)^n * [x^n] (Sum_{k>=0} x^k / (k!)^n)^2. - Ilya Gutkovskiy, Jul 15 2020
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EXAMPLE
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The triangle A209427 of coefficients C(n,k)^n, n>=k>=0, begins:
1;
1, 1;
1, 4, 1;
1, 27, 27, 1;
1, 256, 1296, 256, 1;
1, 3125, 100000, 100000, 3125, 1;
1, 46656, 11390625, 64000000, 11390625, 46656, 1; ...
in which the row sums form this sequence.
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MATHEMATICA
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Table[Sum[Binomial[n, k]^n, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 05 2012 *)
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PROG
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(PARI) a(n)=sum(k=0, n, binomial(n, k)^n)
(Magma) [(&+[Binomial(n, j)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022
(SageMath) [sum(binomial(n, j)^n for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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