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A276559 Expansion of Sum_{k>=1} k^2*x^k^2/(1 - x^k^2) * Product_{k>=1} 1/(1 - x^k^2).

%I A276559 #11 Sep 19 2018 10:48:21
%S A276559 1,2,3,8,10,12,14,24,36,40,44,60,78,84,90,128,153,180,190,240,273,308,
%T A276559 322,384,475,520,567,644,754,810,868,992,1122,1258,1330,1548,1702,
%U A276559 1862,1950,2200,2460,2646,2838,3124,3510,3726,3948,4320,4802,5200,5457,6032,6572,7128,7425,8064,8778,9454,9971,10680
%N A276559 Expansion of Sum_{k>=1} k^2*x^k^2/(1 - x^k^2) * Product_{k>=1} 1/(1 - x^k^2).
%C A276559 Sum of all parts of all partitions of n into squares.
%C A276559 Convolution of the sequences A001156 and A035316.
%H A276559 Alois P. Heinz, <a href="/A276559/b276559.txt">Table of n, a(n) for n = 1..10000</a>
%H A276559 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A276559 G.f.: Sum_{k>=1} k^2*x^k^2/(1 - x^k^2) * Product_{k>=1} 1/(1 - x^k^2).
%F A276559 G.f.: x*f'(x), where f(x) = Product_{k>=1} 1/(1 - x^k^2).
%F A276559 a(n) = n * A001156(n).
%F A276559 a(n) = n * Sum_{k=1..n} A243148(n,k). - _Alois P. Heinz_, Sep 19 2018
%e A276559 a(8) = 24 because we have [4, 4], [4, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1] and 3*8 = 24.
%p A276559 b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], (s->
%p A276559       `if`(s>n, 0, (p->p+[0, p[1]*s])(b(n-s, i))))(i^2)+b(n, i-1))
%p A276559     end:
%p A276559 a:= n-> b(n, isqrt(n))[2]:
%p A276559 seq(a(n), n=1..70);  # _Alois P. Heinz_, Sep 19 2018
%t A276559 nmax = 60; Rest[CoefficientList[Series[Sum[k^2 x^k^2/(1 - x^k^2), {k, 1, nmax}] Product[1/(1 - x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]]
%t A276559 nmax = 60; Rest[CoefficientList[Series[x D[Product[1/(1 - x^k^2), {k, 1, nmax}], x], {x, 0, nmax}], x]]
%Y A276559 Cf. A000290, A001156, A035316, A066186, A243148, A281541.
%K A276559 nonn
%O A276559 1,2
%A A276559 _Ilya Gutkovskiy_, Apr 10 2017