A014787 - OEIS

%I #36 Aug 01 2017 11:50:07

%S 1,12,66,232,627,1452,2982,5544,9669,16016,25158,38160,56266,80124,

%T 111816,153528,205260,270876,353870,452496,574299,724044,895884,

%U 1103520,1353330,1633500,1966482,2360072,2792703,3299340,3892922,4533936,5273841,6134448

%N Expansion of Jacobi theta constant (theta_2/2)^12.

%C Number of ways of writing n as the sum of 12 triangular numbers from A000217.

%H Seiichi Manyama, <a href="/A014787/b014787.txt">Table of n, a(n) for n = 0..10000</a>

%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=12, Theorem 7.

%F From _Wolfdieter Lang_, Jan 13 2017: (Start)

%F G.f.: 12th power of g.f. for A010054.

%F a(n) = (A001160(2*n+3) - A000735(n+1))/256. See the Ono et al. link, case k=12, Theorem 7. (End)

%F a(0) = 1, a(n) = (12/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} 12*(x^k/k)/(1 + x^k)). - _Ilya Gutkovskiy_, Jul 31 2017

%e a(2) = (A001160(7) - A000735(3))/256 = (16808 - (-88))/256 = 66. - _Wolfdieter Lang_, Jan 13 2017

%Y Column k=12 of A286180.

%Y Cf. A000217, A000735, A001160.

%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