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A167009 a(n) = Sum_{k=0..n} C(n^2, n*k). 13
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

V. 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))

CROSSREFS

Cf. A014062, A167010.

Cf. A167006, A209330.

Cf. 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 January 21 12:58 EST 2022. Contains 350477 sequences. (Running on oeis4.)