A167009 a(n) = Sum_{k=0..n} C(n^2, n*k).

1, 2, 8, 170, 16512, 6643782, 11582386286, 79450506979090, 2334899414608412672, 265166261617029717011822, 128442558588779813655233443038, 238431997806538515396060130910954852

Offset: 0

Author

Paul D. Hanna, Nov 17 2009

Keywords

• nonn

Examples

			The triangle A209330 of coefficients C(n^2, n*k), n>=k>=0, begins:
  1;
  1,       1;
  1,       6,          1;
  1,      84,         84,          1;
  1,    1820,      12870,       1820,          1;
  1,   53130,    3268760,    3268760,      53130,       1;
  1, 1947792, 1251677700, 9075135300, 1251677700, 1947792,     1; ...
in which the row sums form this sequence.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..58
• Vaclav Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013.

Crossrefs

Cf. A014062, A167006, A167010, A209330, A218792, A306846.

Programs

Magma

[(&+[Binomial(n^2, n*j): j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022

Mathematica

Table[Sum[Binomial[n^2,n*k],{k,0,n}],{n,0,15}] (* Harvey P. Dale, Dec 11 2011 *)

PARI

a(n)=sum(k=0,n,binomial(n^2,n*k))

Sage

[sum(binomial(n^2, n*j) 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 A167006. - Paul D. Hanna, Nov 18 2009

If n is even then a(n) ~ c * 2^(n^2 + 1/2)/(n*sqrt(Pi)), 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) = A306846(n^2,n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n - x^n) for n > 0. - Seiichi Manyama, Oct 11 2021