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%I #67 Feb 04 2021 15:21:01
%S 1,2,3,5,9,15,23,35,55,87,134,202,305,463,700,1049,1565,2334,3478,
%T 5168,7654,11314,16705,24632,36260,53295,78237,114728,168059,245916,
%U 359483,525021,766144,1117107,1627587,2369609,3447549,5012588,7283577,10577198,15351519,22268890,32286666,46788056,67770831
%N Expansion of Sum_{i>=1} x^(i^2) / (1 - Sum_{j>=1} x^(j^2))^2.
%C Total number of parts in all compositions (ordered partitions) of n into squares (A000290).
%H Alois P. Heinz, <a href="/A281704/b281704.txt">Table of n, a(n) for n = 1..6575</a>
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F G.f.: Sum_{i>=1} x^(i^2) / (1 - Sum_{j>=1} x^(j^2))^2.
%F a(n) = Sum_{k=0..n} k * A337165(n,k). - _Alois P. Heinz_, Feb 03 2021
%e a(6) = 15 because we have [4, 1, 1], [1, 4, 1], [1, 1, 4], [1, 1, 1, 1, 1, 1] and 3 + 3 + 3 + 6 = 15.
%p b:= proc(n) option remember; `if`(n=0, [1, 0], add(
%p (p-> p+[0, p[1]])(b(n-j^2)), j=1..isqrt(n)))
%p end:
%p a:= n-> b(n)[2]:
%p seq(a(n), n=1..45); # _Alois P. Heinz_, Aug 07 2019
%t nmax = 45; Rest[CoefficientList[Series[Sum[x^i^2, {i, 1, nmax}]/(1 - Sum[x^j^2, {j, 1, nmax}])^2, {x, 0, nmax}], x]]
%t nmax = 45; Rest[CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)/(2 (1 + (1 - EllipticTheta[3, 0, x])/2)^2), {x, 0, nmax}], x]]
%Y Cf. A000290, A006456, A337165.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Jan 30 2017
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