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
Vaclav 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 = -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
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)
(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
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Paul D. Hanna, Nov 17 2009
STATUS
approved