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%I #39 Aug 26 2022 17:41:53
%S 1,2,6,56,1810,206252,86874564,132282417920,770670360699138,
%T 16425660314368351892,1367610300690018553312276,
%U 419460465362069257397304825200,509571049488109525160616367158261124,2290638298071684282149128235413262383804352
%N a(n) = Sum_{k=0..n} C(n,k)^n.
%C The number of n*n 0-1 matrices with equal numbers of nonzeros in every row. - _David Eppstein_, Jan 19 2012
%H Vincenzo Librandi, <a href="/A167010/b167010.txt">Table of n, a(n) for n = 0..59</a>
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Interesting asymptotic formulas for binomial sums</a>, Jun 09 2013.
%F Ignoring initial term, equals the logarithmic derivative of A167007. [_Paul D. Hanna_, Nov 18 2009]
%F 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
%F If n is odd then c = Sum_{k = -infinity..infinity} exp(-2*(k+1/2)^2) = 1.23528676585389... - _Vaclav Kotesovec_, Nov 06 2012
%F a(n) = (n!)^n * [x^n] (Sum_{k>=0} x^k / (k!)^n)^2. - _Ilya Gutkovskiy_, Jul 15 2020
%e The triangle A209427 of coefficients C(n,k)^n, n>=k>=0, begins:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 27, 27, 1;
%e 1, 256, 1296, 256, 1;
%e 1, 3125, 100000, 100000, 3125, 1;
%e 1, 46656, 11390625, 64000000, 11390625, 46656, 1; ...
%e in which the row sums form this sequence.
%t Table[Sum[Binomial[n, k]^n, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 05 2012 *)
%o (PARI) a(n)=sum(k=0,n,binomial(n,k)^n)
%o (Magma) [(&+[Binomial(n,j)^n: j in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Aug 26 2022
%o (SageMath) [sum(binomial(n,j)^n for j in (0..n)) for n in (0..20)] # _G. C. Greubel_, Aug 26 2022
%Y Cf. A000312, A014062, A066300, A167009, A167007, A209427, A218792.
%K nonn,nice
%O 0,2
%A _Paul D. Hanna_, Nov 17 2009
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