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A167009 a(n) = Sum_{k=0..n} C(n^2, n*k). 14
1, 2, 8, 170, 16512, 6643782, 11582386286, 79450506979090, 2334899414608412672, 265166261617029717011822, 128442558588779813655233443038, 238431997806538515396060130910954852 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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
Sequence in context: A009606 A009682 A076548 * A350808 A009713 A265597
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 17 2009
STATUS
approved

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Last modified February 29 14:32 EST 2024. Contains 370425 sequences. (Running on oeis4.)