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%I #49 Aug 01 2017 11:50:14
%S 1,24,276,2048,11178,48576,177400,565248,1612875,4200352,10131156,
%T 22892544,48897678,99448320,193740408,363315200,658523925,1157743824,
%U 1980143600,3303168000,5386270686,8602175744,13477895856,20748607488,31425764410,46883528256,68969957700
%N Expansion of Jacobi theta constant (theta_2/2)^24.
%C Number of ways of writing n as the sum of 24 triangular numbers from A000217.
%H Seiichi Manyama, <a href="/A014809/b014809.txt">Table of n, a(n) for n = 0..10000</a>
%H J. G. Huard and K. S. Williams, <a href="https://doi.org/10.1216/rmjm/1181069710">Sums of sixteen and twenty-four triangular numbers</a>, Rocky Mountain J. Math. Volume 35, Number 3 (2005), 857-868.
%H K. Ono, S. Robins and P. T. Wahl, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/006.pdf">On the representation of integers as sums of triangular numbers</a>, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Case k=24, Theorem 8.
%F From _Wolfdieter Lang_, Jan 13 2017: (Start)
%F G.f.: 24th power of the g.f. for A010054.
%F a(n) = (A096963(n+3) - tau(n+3) - 2072*tau((n+3)/2))/176896, with Ramanujan's tau function given in A000594, and tau(n) is put to 0 if n is not integer. See the Ono et al. link, case k=24, Theorem 8.
%F (End)
%F a(n) = 1/72 * Sum_{a, b, x, y > 0, a*x + b*y = n + 3, x == y == 1 mod 2 and a > b} (a*b)^3*(a^2 - b^2)^2. - _Seiichi Manyama_, May 05 2017
%F a(0) = 1, a(n) = (24/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - _Seiichi Manyama_, May 06 2017
%F G.f.: exp(Sum_{k>=1} 24*(x^k/k)/(1 + x^k)). - _Ilya Gutkovskiy_, Jul 31 2017
%Y Column k=24 of A286180.
%Y Cf. A000217, A000594, A096963.
%Y Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Seiichi Manyama_, May 05 2017
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