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A302833 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^2)).

%I A302833 #12 Apr 14 2018 14:42:48
%S A302833 1,2,3,4,6,8,10,12,15,19,23,27,32,38,44,50,58,67,77,87,99,112,126,140,
%T A302833 156,175,195,216,239,265,292,320,351,385,422,460,503,549,598,648,703,
%U A302833 763,826,892,963,1041,1122,1206,1296,1394,1498,1605,1721,1845,1977,2112,2256,2410,2573
%N A302833 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^2)).
%C A302833 Partial sums of A001156.
%C A302833 Number of partitions of n into squares if there are two kinds of 1's.
%H A302833 Alois P. Heinz, <a href="/A302833/b302833.txt">Table of n, a(n) for n = 0..20000</a>
%H A302833 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A302833 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A302833 G.f.: (1/(1 - x))*Sum_{j>=0} x^(j^2)/Product_{k=1..j} (1 - x^(k^2)).
%F A302833 From _Vaclav Kotesovec_, Apr 13 2018: (Start)
%F A302833 a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3)) / (2*Pi^(3/2) * sqrt(3*n)).
%F A302833 a(n) ~ 2^(4/3) * n^(2/3) / (Pi^(1/3) * Zeta(3/2)^(2/3)) * A001156(n). (End)
%p A302833 b:= proc(n, i) option remember; `if`(n=0 or i=1, n+1,
%p A302833       b(n, i-1)+ `if`(i^2>n, 0, b(n-i^2, i)))
%p A302833     end:
%p A302833 a:= n-> b(n, isqrt(n)):
%p A302833 seq(a(n), n=0..100);  # _Alois P. Heinz_, Apr 13 2018
%t A302833 nmax = 58; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]
%t A302833 nmax = 58; CoefficientList[Series[1/(1 - x) Sum[x^j^2/Product[(1 - x^k^2), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]
%Y A302833 Cf. A000070, A000290, A001156, A078134, A279225, A302835.
%K A302833 nonn
%O A302833 0,2
%A A302833 _Ilya Gutkovskiy_, Apr 13 2018