A338464 Number of ways to write 2*n as an ordered sum of n squares of positive integers.

1, 0, 0, 3, 0, 0, 15, 0, 8, 84, 0, 110, 495, 0, 1092, 3018, 120, 9520, 18870, 2907, 77520, 120270, 43890, 606188, 780023, 531300, 4620200, 5161377, 5651100, 34622172, 35045340, 55234560, 256503672, 245772464, 508930224, 1886151225, 1788167610, 4491607230

Offset: 0

Author

Ilya Gutkovskiy, Jan 31 2021

Keywords

• nonn

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

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

Programs

Maple

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..39);  # Alois P. Heinz, Feb 04 2021

Mathematica

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

Formula

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

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

a(n) = A337165(2n,n). - Alois P. Heinz, Feb 04 2021