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A282209 Expansion of Product_{k>=1} 1/(1 - k^2*x^(k^2)).

%I A282209 #8 Feb 09 2017 11:13:39
%S A282209 1,1,1,1,5,5,5,5,21,30,30,30,94,130,130,130,402,546,627,627,1715,2291,
%T A282209 2615,2615,6967,9440,10736,11465,28873,38765,43949,46865,116753,
%U A282209 156321,178578,190242,476391,634663,723691,770347,1914943,2550735,2906847,3107160,7685544
%N A282209 Expansion of Product_{k>=1} 1/(1 - k^2*x^(k^2)).
%C A282209 Sum of products of terms in all partitions of n into squares (A000290).
%H A282209 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A282209 G.f.: Product_{k>=1} 1/(1 - k^2*x^(k^2)).
%F A282209 From _Vaclav Kotesovec_, Feb 09 2017: (Start)
%F A282209 a(n) ~ c * 2^(n/2), where:
%F A282209 c = 1.84902025727376837058629436557644856279088... if n == 0 (mod 4),
%F A282209 c = 1.74739571210218418633067606853005648684028... if n == 1 (mod 4),
%F A282209 c = 1.41060067910504703778072732362810764186990... if n == 2 (mod 4),
%F A282209 c = 1.06705333199321743850009229910087278853310... if n == 3 (mod 4).
%F A282209 (End)
%e A282209 a(8) = 21 because we have [4, 4], [4, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1], 4*4 = 16, 4*1*1*1*1 = 4, 1*1*1*1*1*1*1*1 = 1 and 16 + 4 + 1 = 21.
%t A282209 nmax = 44; CoefficientList[Series[Product[1/(1 - k^2 x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A282209 Cf. A000290, A001156, A006906.
%K A282209 nonn
%O A282209 0,5
%A A282209 _Ilya Gutkovskiy_, Feb 09 2017