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%I #31 Aug 02 2017 10:22:45
%S 1,5,10,15,25,31,35,55,60,60,90,90,95,135,125,126,170,180,175,215,220,
%T 195,285,280,245,340,300,320,405,350,351,450,465,415,515,480,425,620,
%U 590,505,655,625,590,755,660,650,805,770,755,855,841,730,1045,960,770,1100
%N Expansion of Jacobi theta constant theta_2^5 /32.
%C Also number of ways of writing n as a sum of five triangular numbers. - _N. J. A. Sloane_, Jun 01 2013
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
%H Seiichi Manyama, <a href="/A008439/b008439.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.
%F a(0) = 1, a(n) = (5/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} 5*(x^k/k)/(1 + x^k)). - _Ilya Gutkovskiy_, Jul 31 2017
%t a002129[n_]:=-Sum[(-1)^d*d, {d, Divisors[n]}]; a[n_]:=a[n]=If[n==0, 1, 5 Sum[a002129[k] a[n - k], {k, n}]/n]; Table[a[n], {n, 0, 100}] (* _Indranil Ghosh_, Aug 02 2017 *)
%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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