A298329 Number of ordered ways of writing n^2 as a sum of n squares of nonnegative integers.

1, 1, 2, 6, 5, 90, 582, 4081, 45678, 378049, 3844532, 39039539, 395170118, 4589810849, 53154371025, 660113986997, 8584476248237, 113555197832758, 1572878837435750, 22259911738401660, 324143769099772448, 4869443438412466557, 74837370448784241452, 1182177603062005007658

Offset: 0

Author

Ilya Gutkovskiy, Jan 17 2018

Keywords

• nonn

Examples

			a(3) = 6 because we have [9, 0, 0], [4, 4, 1], [4, 1, 4], [1, 4, 4], [0, 9, 0] and [0, 0, 9].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100 (first 81 terms from Seiichi Manyama)
• Index entries for sequences related to sums of squares

Crossrefs

[x^(n^b)] (Sum_{k>=0} x^(k^b))^n: A088218 (b=1), this sequence (b=2), A298671 (b=3).

Cf. A037444, A066535, A232173, A287617, A298330, A300446.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1/t!, (s->
     `if`(s*t n!*b(n^2, n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 28 2018

Mathematica

Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^2}], {n, 0, 23}]

PARI

{a(n) = polcoeff((sum(k=0, n, x^(k^2)+x*O(x^(n^2))))^n, n^2)} \\ Seiichi Manyama, Oct 28 2018

Formula

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