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A298329 Number of ordered ways of writing n^2 as a sum of n squares of nonnegative integers. 12

%I

%S 1,1,2,6,5,90,582,4081,45678,378049,3844532,39039539,395170118,

%T 4589810849,53154371025,660113986997,8584476248237,113555197832758,

%U 1572878837435750,22259911738401660,324143769099772448,4869443438412466557,74837370448784241452,1182177603062005007658

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

%H Alois P. Heinz, <a href="/A298329/b298329.txt">Table of n, a(n) for n = 0..100</a> (first 81 terms from Seiichi Manyama)

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

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

%e 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].

%p b:= proc(n, i, t) option remember; `if`(n=0, 1/t!, (s->

%p `if`(s*t<n, 0, add(b(n-s*j, i-1, t-j)/j!, j=0..min(t, n/s))))(i^2))

%p end:

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

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Oct 28 2018

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

%o (PARI) {a(n) = polcoeff((sum(k=0, n, x^(k^2)+x*O(x^(n^2))))^n, n^2)} \\ _Seiichi Manyama_, Oct 28 2018

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

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jan 17 2018

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Last modified May 30 18:05 EDT 2020. Contains 334728 sequences. (Running on oeis4.)