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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 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 Cf. A014062, A167006, A167010, A209330, A218792, A306846. Sequence in context: A009606 A009682 A076548 * A350808 A009713 A265597 Adjacent sequences: A167006 A167007 A167008 * A167010 A167011 A167012 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.)