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 A320932 a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2). 6

%I

%S 1,1,1,2,2,6,20,51,141,381,1001,2796,7861,22306,64129,185692,540468,

%T 1585246,4674464,13846636,41216933,123176849,369410571,1111661833,

%U 3355466306,10156304314,30821794651,93761053797,285859742756,873355481467,2673455511946,8198687383812

%N a(n) = [x^(n*(n+1)/2)] 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*(n+1)/2.

%H Alois P. Heinz, <a href="/A320932/b320932.txt">Table of n, a(n) for n = 0..300</a> (first 101 terms from Seiichi Manyama)

%F a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function.

%e 1*1^2 + 2*1^2 + 3*1^2 + 4*1^2 + 5*1^2 = 15.

%e 1*2^2 + 2*1^2 + 3*0^2 + 4*1^2 + 5*1^2 = 15.

%e 1*0^2 + 2*2^2 + 3*1^2 + 4*1^2 + 5*0^2 = 15.

%e 1*3^2 + 2*1^2 + 3*0^2 + 4*1^2 + 5*0^2 = 15.

%e 1*1^2 + 2*1^2 + 3*2^2 + 4*0^2 + 5*0^2 = 15.

%e 1*2^2 + 2*2^2 + 3*1^2 + 4*0^2 + 5*0^2 = 15.

%e So a(5) = 6.

%p b:= proc(n, i) option remember; local j; if n=0 then 1

%p elif i<1 then 0 else b(n, i-1); for j while

%p i*j^2<=n do %+b(n-i*j^2, i-1) od; % fi

%p end:

%p a:= n-> b(n*(n+1)/2, n):

%p seq(a(n), n=0..40); # _Alois P. Heinz_, Oct 28 2018

%t nmax = 30; Table[SeriesCoefficient[Product[(EllipticTheta[3, 0, x^k] + 1)/2, {k, 1, n}], {x, 0, n*(n+1)/2}], {n, 0, nmax}] (* _Vaclav Kotesovec_, Oct 29 2018 *)

%o (PARI) {a(n) = polcoeff(prod(i=1, n, sum(j=0, sqrtint(n*(n+1)\(2*i)), x^(i*j^2)+x*O(x^(n*(n+1)/2)))), n*(n+1)/2)}

%Y Cf. A000122, A000217, A010052, A173519, A300446, A320931.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Oct 28 2018

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Last modified May 8 19:28 EDT 2021. Contains 343666 sequences. (Running on oeis4.)