A338464 - OEIS

%I #11 Feb 04 2021 15:22:12

%S 1,0,0,3,0,0,15,0,8,84,0,110,495,0,1092,3018,120,9520,18870,2907,

%T 77520,120270,43890,606188,780023,531300,4620200,5161377,5651100,

%U 34622172,35045340,55234560,256503672,245772464,508930224,1886151225,1788167610,4491607230

%N Number of ways to write 2*n as an ordered sum of n squares of positive integers.

%C Also number of ways to write n as an ordered sum of n nonnegative numbers one less than a square.

%H Alois P. Heinz, <a href="/A338464/b338464.txt">Table of n, a(n) for n = 0..2000</a>

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

%F a(n) = [x^n] (Sum_{k>=0} x^(k*(k + 2)))^n.

%F a(n) = A337165(2n,n). - _Alois P. Heinz_, Feb 04 2021

%p b:= proc(n, t) option remember;

%p       `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->

%p       `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))

%p     end:

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

%p seq(a(n), n=0..39);  # _Alois P. Heinz_, Feb 04 2021

%t Table[SeriesCoefficient[((EllipticTheta[3, 0, x] - 1)/2)^n, {x, 0, 2 n}], {n, 0, 37}]

%Y Cf. A000290, A005563, A066535, A111178, A287617, A303333, A337165.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Jan 31 2021