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%I #9 Nov 01 2018 12:22:34
%S 1,1,1,4,16,87,911,8081,82494,1108584,14559487,206462480,3300362073,
%T 54235076625,939612600043,17366394088532,332129019947772,
%U 6615538793829307,137564490944940832,2954281836759475893,65572183746807351880,1503752010271535590284,35476544827929325305961
%N a(n) = [x^(n^3)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).
%C Also the number of nonnegative integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n^3.
%e 1*0^2 + 2*0^2 + 3*0^2 + 4*4^2 = 64.
%e 1*0^2 + 2*0^2 + 3*4^2 + 4*2^2 = 64.
%e 1*1^2 + 2*0^2 + 3*3^2 + 4*3^2 = 64.
%e 1*1^2 + 2*4^2 + 3*3^2 + 4*1^2 = 64.
%e 1*2^2 + 2*2^2 + 3*4^2 + 4*1^2 = 64.
%e 1*2^2 + 2*4^2 + 3*2^2 + 4*2^2 = 64.
%e 1*4^2 + 2*0^2 + 3*2^2 + 4*3^2 = 64.
%e 1*4^2 + 2*0^2 + 3*4^2 + 4*0^2 = 64.
%e 1*4^2 + 2*4^2 + 3*0^2 + 4*2^2 = 64.
%e 1*4^2 + 2*4^2 + 3*2^2 + 4*1^2 = 64.
%e 1*5^2 + 2*0^2 + 3*1^2 + 4*3^2 = 64.
%e 1*5^2 + 2*2^2 + 3*3^2 + 4*1^2 = 64.
%e 1*5^2 + 2*4^2 + 3*1^2 + 4*1^2 = 64.
%e 1*6^2 + 2*0^2 + 3*2^2 + 4*2^2 = 64.
%e 1*7^2 + 2*2^2 + 3*1^2 + 4*1^2 = 64.
%e 1*8^2 + 2*0^2 + 3*0^2 + 4*0^2 = 64.
%e So a(4) = 16.
%o (PARI) {a(n) = polcoeff(prod(i=1, n, sum(j=0, sqrtint(n^3\i), x^(i*j^2)+x*O(x^(n^3)))), n^3)}
%Y Cf. A300446, A321139, A321239.
%K nonn
%O 0,4
%A _Seiichi Manyama_, Nov 01 2018
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