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%I #17 Feb 03 2024 10:13:52
%S 1,1,3,16,141,1534,19111,262103,3853373,59763670,966945204,
%T 16191250596,278933800080,4921604827876,88627915588351,
%U 1624349874930925,30231112607904743,570284342486800214,10887435073866747752,210086404047975194316,4092940691144348506396,80432925119259253535963
%N a(n) = [x^(n^3)] Product_{k=1..n} Sum_{m>=0} x^(k^2*m).
%C Also the number of nonnegative integer solutions (a_1, a_2, ..., a_n) to the equation 1^2*a_1 + 2^2*a_2 + ... + n^2*a_n = n^3.
%C Also the number of partitions of n^3 into square parts not greater than n^2. - _Paul D. Hanna_, Feb 02 2024
%e 1^2* 0 + 2^2*0 + 3^2*3 = 27.
%e 1^2* 1 + 2^2*2 + 3^2*2 = 27.
%e 1^2* 2 + 2^2*4 + 3^2*1 = 27.
%e 1^2* 3 + 2^2*6 + 3^2*0 = 27.
%e 1^2* 5 + 2^2*1 + 3^2*2 = 27.
%e 1^2* 6 + 2^2*3 + 3^2*1 = 27.
%e 1^2* 7 + 2^2*5 + 3^2*0 = 27.
%e 1^2* 9 + 2^2*0 + 3^2*2 = 27.
%e 1^2*10 + 2^2*2 + 3^2*1 = 27.
%e 1^2*11 + 2^2*4 + 3^2*0 = 27.
%e 1^2*14 + 2^2*1 + 3^2*1 = 27.
%e 1^2*15 + 2^2*3 + 3^2*0 = 27.
%e 1^2*18 + 2^2*0 + 3^2*1 = 27.
%e 1^2*19 + 2^2*2 + 3^2*0 = 27.
%e 1^2*23 + 2^2*1 + 3^2*0 = 27.
%e 1^2*27 + 2^2*0 + 3^2*0 = 27.
%e So a(3) = 16.
%o (PARI) {a(n) = polcoeff(prod(i=1, n, sum(j=0, n^3\i^2, x^(i^2*j)+x*O(x^(n^3)))), n^3)}
%o (PARI) {a(n) = polcoeff( 1/prod(k=1,n, 1 - x^(k^2) +x*O(x^(n^3)) ), n^3) }
%o for(n=0,20, print1(a(n),", ")) \\ _Paul D. Hanna_, Feb 02 2024
%Y Cf. A001156, A107379, A321238.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Nov 01 2018
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