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A291649 Expansion of Product_{k>=1} (1 + x^(k^2))^(k^2).

%I A291649 #20 Aug 28 2018 13:05:15
%S A291649 1,1,0,0,4,4,0,0,6,15,9,0,4,40,36,0,17,71,90,36,64,100,180,144,96,274,
%T A291649 394,300,148,740,820,480,472,1150,1851,1341,1146,1318,3880,3540,1704,
%U A291649 3017,6455,7134,3780,7822,9574,12180,10304,12057,19750,22485,20558,15910,43076,43236,31104,33742,66895
%N A291649 Expansion of Product_{k>=1} (1 + x^(k^2))^(k^2).
%C A291649 Number of partitions of n into distinct squares, where k^2 different parts of size k^2 are available (1a, 4a, 4b, 4c, 4d, ...).
%H A291649 Vaclav Kotesovec, <a href="/A291649/b291649.txt">Table of n, a(n) for n = 0..10000</a>
%H A291649 <a href="/index/Par#part">Index entries for related partition-counting sequences</a>
%F A291649 G.f.: Product_{k>=1} (1 + x^A000290(k))^A000290(k).
%F A291649 a(n) ~ exp(5 * 2^(-9/5) * 3^(-3/5) * (9-4*sqrt(2))^(1/5) * Pi^(1/5) * Zeta(5/2)^(2/5) * n^(3/5)) * 3^(1/5) * (2*sqrt(2)-1)^(1/5) * Zeta(5/2)^(1/5) / (2^(9/10) * sqrt(5) * Pi^(2/5) * n^(7/10)). - _Vaclav Kotesovec_, Aug 29 2017
%e A291649 a(8) = 6 because we have [4a, 4b], [4a, 4c], [4a, 4d], [4b, 4c], [4b, 4d] and [4c, 4d].
%t A291649 nmax = 100; CoefficientList[Series[Product[(1 + x^k^2)^k^2, {k, 1, nmax}], {x, 0, nmax}], x]
%t A291649 nmax = 100; s = 1 + x; Do[s *= Sum[Binomial[k^2, j]*x^(j*k^2), {j, 0, Floor[nmax/k^2] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x] (* _Vaclav Kotesovec_, Aug 28 2017 *)
%Y A291649 Cf. A000290, A026007, A027998, A033461, A262736, A281790, A284896, A291655, A291692.
%K A291649 nonn
%O A291649 0,5
%A A291649 _Ilya Gutkovskiy_, Aug 28 2017