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A167010 a(n) = Sum_{k=0..n} C(n,k)^n. 23
1, 2, 6, 56, 1810, 206252, 86874564, 132282417920, 770670360699138, 16425660314368351892, 1367610300690018553312276, 419460465362069257397304825200, 509571049488109525160616367158261124, 2290638298071684282149128235413262383804352 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..59

V. Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013

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 = -infinity..infinity} 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

EXAMPLE

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.

MATHEMATICA

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

PROG

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

CROSSREFS

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

Cf. A218792.

Sequence in context: A000146 A318001 A211933 * A014070 A320287 A198445

Adjacent sequences:  A167007 A167008 A167009 * A167011 A167012 A167013

KEYWORD

nonn,nice

AUTHOR

Paul D. Hanna, Nov 17 2009

STATUS

approved

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Last modified November 27 23:47 EST 2020. Contains 338685 sequences. (Running on oeis4.)