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A014787
Expansion of Jacobi theta constant (theta_2/2)^12.
16
1, 12, 66, 232, 627, 1452, 2982, 5544, 9669, 16016, 25158, 38160, 56266, 80124, 111816, 153528, 205260, 270876, 353870, 452496, 574299, 724044, 895884, 1103520, 1353330, 1633500, 1966482, 2360072, 2792703, 3299340, 3892922, 4533936, 5273841, 6134448
OFFSET
0,2
COMMENTS
Number of ways of writing n as the sum of 12 triangular numbers from A000217.
LINKS
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=12, Theorem 7.
FORMULA
From Wolfdieter Lang, Jan 13 2017: (Start)
G.f.: 12th power of g.f. for A010054.
a(n) = (A001160(2*n+3) - A000735(n+1))/256. See the Ono et al. link, case k=12, Theorem 7. (End)
a(0) = 1, a(n) = (12/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 12*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/256) * exp(3*Pi/2) * Pi^3 * sqrt(2) / Gamma(3/4)^12 = A388241. - Simon Plouffe, Sep 15 2025
EXAMPLE
a(2) = (A001160(7) - A000735(3))/256 = (16808 - (-88))/256 = 66. - Wolfdieter Lang, Jan 13 2017
CROSSREFS
Column k=12 of A286180.
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: A226235 A045853 A277104 * A007249 A112142 A271870
KEYWORD
nonn
EXTENSIONS
More terms from Seiichi Manyama, May 05 2017
STATUS
approved