A167010 a(n) = Sum_{k=0..n} C(n,k)^n.

1, 2, 6, 56, 1810, 206252, 86874564, 132282417920, 770670360699138, 16425660314368351892, 1367610300690018553312276, 419460465362069257397304825200, 509571049488109525160616367158261124, 2290638298071684282149128235413262383804352

Offset: 0

Author

Paul D. Hanna, Nov 17 2009

Keywords

• nonn
• nice

Comments

The number of n*n 0-1 matrices with equal numbers of nonzeros in every row. - David Eppstein, Jan 19 2012

Examples

			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.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..59
• Vaclav Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013.

Crossrefs

Cf. A000312, A014062, A066300, A167009, A167007, A209427, A218792.

Programs

Magma

[(&+[Binomial(n,j)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022

Mathematica

Table[Sum[Binomial[n, k]^n, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 05 2012 *)

PARI

a(n)=sum(k=0,n,binomial(n,k)^n)

SageMath

[sum(binomial(n,j)^n for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Formula

Ignoring initial term, equals the logarithmic derivative of A167007. [Paul D. Hanna, Nov 18 2009]

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