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%I #7 Aug 07 2019 18:40:53
%S 1,2,4,8,14,25,45,77,131,224,377,629,1049,1738,2863,4708,7716,12598,
%T 20524,33363,54102,87567,141489,228216,367538,591098,949372,1522917,
%U 2440190,3905747,6245198,9976535,15923083,25392755,40462155,64426278,102510580,162997910,259010672,411328655,652842792,1035591110
%N Expansion of Sum_{i>=1} x^(i*(i+1)/2) / (1 - Sum_{j>=1} x^(j*(j+1)/2))^2.
%C Total number of parts in all compositions (ordered partitions) of n into nonzero triangular numbers (A000217).
%H Alois P. Heinz, <a href="/A281810/b281810.txt">Table of n, a(n) for n = 1..5238</a>
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F G.f.: Sum_{i>=1} x^(i*(i+1)/2) / (1 - Sum_{j>=1} x^(j*(j+1)/2))^2.
%e a(6) = 25 because we have [6], [3, 3], [3, 1, 1, 1], [1, 3, 1, 1], [1, 1, 3, 1], [1, 1, 1, 3], [1, 1, 1, 1, 1, 1] and 1 + 2 + 4 + 4 + 4 + 4 + 6 = 25.
%p b:= proc(n) option remember; `if`(n=0, [1, 0], add(
%p (p-> p+[0, p[1]])(b(n-j*(j+1)/2)), j=1..isqrt(2*n)))
%p end:
%p a:= n-> b(n)[2]:
%p seq(a(n), n=1..55); # _Alois P. Heinz_, Aug 07 2019
%t nmax = 42; Rest[CoefficientList[Series[Sum[x^(i (i + 1)/2), {i, 1, nmax}]/(1 - Sum[x^(j (j + 1)/2), {j, 1, nmax}])^2, {x, 0, nmax}], x]]
%t nmax = 42; Rest[CoefficientList[Series[(2 x^(1/8) EllipticTheta[2, 0, Sqrt[x]] - 4 x^(1/4))/(4 x^(1/8) - EllipticTheta[2, 0, Sqrt[x]])^2, {x, 0, nmax}], x]]
%Y Cf. A000217, A023361.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Jan 30 2017
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