A282209 - OEIS

%I #8 Feb 09 2017 11:13:39

%S 1,1,1,1,5,5,5,5,21,30,30,30,94,130,130,130,402,546,627,627,1715,2291,

%T 2615,2615,6967,9440,10736,11465,28873,38765,43949,46865,116753,

%U 156321,178578,190242,476391,634663,723691,770347,1914943,2550735,2906847,3107160,7685544

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

%C Sum of products of terms in all partitions of n into squares (A000290).

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%F G.f.: Product_{k>=1} 1/(1 - k^2*x^(k^2)).

%F From _Vaclav Kotesovec_, Feb 09 2017: (Start)

%F a(n) ~ c * 2^(n/2), where:

%F c = 1.84902025727376837058629436557644856279088... if n == 0 (mod 4),

%F c = 1.74739571210218418633067606853005648684028... if n == 1 (mod 4),

%F c = 1.41060067910504703778072732362810764186990... if n == 2 (mod 4),

%F c = 1.06705333199321743850009229910087278853310... if n == 3 (mod 4).

%F (End)

%e 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 nmax = 44; CoefficientList[Series[Product[1/(1 - k^2 x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A000290, A001156, A006906.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Feb 09 2017