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

0,3

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

Formula

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

Example

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].

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1/t!, (s->
     `if`(s*t<n, 0, add(b(n-s*j, i-1, t-j)/j!, j=0..min(t, n/s))))(i^2))
    end:
a:= n-> 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}]

Prog

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

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.

Sequence in context: A175350 A085057 A069113 * A009462 A346712 A111119

Adjacent sequences: A298326 A298327 A298328 * A298330 A298331 A298332

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 17 2018

Status

approved