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A321139 a(n) = [x^(n^2)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).

%I A321139 #25 Oct 29 2018 12:43:30
%S A321139 1,1,1,3,7,17,52,144,480,1732,5902,21078,78434,289107,1079949,4094643,
%T A321139 15574377,59667023,230318968,892694240,3477119540,13606993083,
%U A321139 53438614380,210622413188,832922044686,3303392730698,13137474884294,52381331536536,209340904575968
%N A321139 a(n) = [x^(n^2)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).
%C A321139 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^2.
%H A321139 Alois P. Heinz, <a href="/A321139/b321139.txt">Table of n, a(n) for n = 0..250</a> (first 101 terms from Seiichi Manyama)
%F A321139 a(n) = [x^(n^2)] Product_{k=1..n} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function.
%e A321139 1*0^2 + 2*1^2 + 3*1^2 + 4*0^2 + 5*2^2 = 25.
%e A321139 1*0^2 + 2*2^2 + 3*2^2 + 4*0^2 + 5*1^2 = 25.
%e A321139 1*0^2 + 2*3^2 + 3*1^2 + 4*1^2 + 5*0^2 = 25.
%e A321139 1*1^2 + 2*0^2 + 3*0^2 + 4*1^2 + 5*2^2 = 25.
%e A321139 1*1^2 + 2*0^2 + 3*1^2 + 4*2^2 + 5*1^2 = 25.
%e A321139 1*1^2 + 2*2^2 + 3*0^2 + 4*2^2 + 5*0^2 = 25.
%e A321139 1*1^2 + 2*2^2 + 3*2^2 + 4*1^2 + 5*0^2 = 25.
%e A321139 1*2^2 + 2*0^2 + 3*0^2 + 4*2^2 + 5*1^2 = 25.
%e A321139 1*2^2 + 2*0^2 + 3*2^2 + 4*1^2 + 5*1^2 = 25.
%e A321139 1*2^2 + 2*1^2 + 3*1^2 + 4*2^2 + 5*0^2 = 25.
%e A321139 1*2^2 + 2*3^2 + 3*1^2 + 4*0^2 + 5*0^2 = 25.
%e A321139 1*3^2 + 2*0^2 + 3*0^2 + 4*2^2 + 5*0^2 = 25.
%e A321139 1*3^2 + 2*0^2 + 3*2^2 + 4*1^2 + 5*0^2 = 25.
%e A321139 1*3^2 + 2*2^2 + 3*1^2 + 4*0^2 + 5*1^2 = 25.
%e A321139 1*4^2 + 2*0^2 + 3*0^2 + 4*1^2 + 5*1^2 = 25.
%e A321139 1*4^2 + 2*1^2 + 3*1^2 + 4*1^2 + 5*0^2 = 25.
%e A321139 1*5^2 + 2*0^2 + 3*0^2 + 4*0^2 + 5*0^2 = 25.
%e A321139 So a(5) = 17.
%p A321139 b:= proc(n, i) option remember; local j; if n=0 then 1
%p A321139       elif i<1 then 0 else b(n, i-1); for j while
%p A321139         i*j^2<=n do %+b(n-i*j^2, i-1) od; % fi
%p A321139     end:
%p A321139 a:= n-> b(n^2, n):
%p A321139 seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 28 2018
%t A321139 nmax = 25; Table[SeriesCoefficient[Product[(EllipticTheta[3, 0, x^k] + 1)/2, {k, 1, n}], {x, 0, n^2}], {n, 0, nmax}] (* _Vaclav Kotesovec_, Oct 29 2018 *)
%o A321139 (PARI) {a(n) = polcoeff(prod(i=1, n, sum(j=0, sqrtint(n^2\i), x^(i*j^2)+x*O(x^(n^2)))), n^2)}
%Y A321139 Cf. A000122, A010052, A206226, A300446, A320932.
%K A321139 nonn
%O A321139 0,4
%A A321139 _Seiichi Manyama_, Oct 28 2018