OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..58
Vaclav Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013.
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
EXAMPLE
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.
MATHEMATICA
Table[Sum[Binomial[n^2, n*k], {k, 0, n}], {n, 0, 15}] (* Harvey P. Dale, Dec 11 2011 *)
PROG
(PARI) a(n)=sum(k=0, n, binomial(n^2, n*k))
(Magma) [(&+[Binomial(n^2, n*j): j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022
(Sage) [sum(binomial(n^2, n*j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 17 2009
STATUS
approved