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

0,4

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

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

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}]

Crossrefs

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

Sequence in context: A208848 A277945 A169777 * A278208 A105786 A291802

Adjacent sequences: A338461 A338462 A338463 * A338465 A338466 A338467

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 31 2021

Status

approved