A014809 Expansion of Jacobi theta constant (theta_2/2)^24.

1, 24, 276, 2048, 11178, 48576, 177400, 565248, 1612875, 4200352, 10131156, 22892544, 48897678, 99448320, 193740408, 363315200, 658523925, 1157743824, 1980143600, 3303168000, 5386270686, 8602175744, 13477895856, 20748607488, 31425764410, 46883528256, 68969957700

Offset

0,2

Comments

Number of ways of writing n as the sum of 24 triangular numbers from A000217.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

J. G. Huard and K. S. Williams, Sums of sixteen and twenty-four triangular numbers, Rocky Mountain J. Math. Volume 35, Number 3 (2005), 857-868.

K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Case k=24, Theorem 8.

Formula

From Wolfdieter Lang, Jan 13 2017: (Start)

G.f.: 24th power of the g.f. for A010054.

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.

(End)

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

a(0) = 1, a(n) = (24/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017

G.f.: exp(Sum_{k>=1} 24*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

Crossrefs

Column k=24 of A286180.

Cf. A000217, A000594, A096963.

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.

Sequence in context: A006665 A010940 A045854 * A007191 A097340 A222156

Adjacent sequences: A014806 A014807 A014808 * A014810 A014811 A014812

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Seiichi Manyama, May 05 2017

Status

approved