A232173 Number of ways of writing n^2 as a sum of n squares.

1, 2, 4, 30, 24, 1210, 18396, 235998, 4793456, 76168850, 1282320348, 25100418046, 481341997032, 10452086347274, 237925595533164, 5524220670435982, 136705837928870368, 3444192369181374754, 89772662325079950436, 2431910317560215089758, 67517711482300160612104

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200 (first 101 terms from Paul D. Hanna)

Formula

a(n) equals the coefficient of x^(n^2) in the n-th power of Jacobi theta_3(x) where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Example

There are a(4) = 24 solutions (w,x,y,z) of 4^2 = w^2 + x^2 + y^2 + z^2:
(2,2,2,2), (-2,-2,-2,-2), 6 permutations of (2,2,-2,-2),
4 permutations of (2,2,2,-2), 4 permutations of (2,-2,-2,-2),
4 permutations of (4,0,0,0), and 4 permutations of (-4,0,0,0).
To illustrate a(n) = the coefficient of x^(n^2) in theta_3(x)^n, where
theta_3(x) = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + 2*x^36 + 2*x^49 +...,
form a table of coefficients of x^k in theta_3(x)^n, n>=0, like so:
n\k:0..1...2...3...4...5...6...7...8...9..10..11..12..13..14..15..16....
0:[(1),0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,...];
1: [1,(2), 0,  0,  2,  0,  0,  0,  0,  2,  0,  0,  0,  0,  0,  0,  2,...];
2: [1, 4,  4,  0, (4), 8,  0,  0,  4,  4,  8,  0,  0,  8,  0,  0,  4,...];
3: [1, 6, 12,  8,  6, 24, 24,  0, 12,(30),24, 24,  8, 24, 48,  0,  6,...];
4: [1, 8, 24, 32, 24, 48, 96, 64, 24,104,144, 96, 96,112,192,192,(24),...];
5: [1,10, 40, 80, 90,112,240,320,200,250,560,560,400,560,800,960,730,...];
then the coefficients in parenthesis form the initial terms of this sequence.

Maple

b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
      b(n, t-1) +2*add(b(n-j^2, t-1), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 10 2023

Mathematica

b[n_, t_] := b[n, t] = If[n == 0, 1, If[n < 0 || t < 1, 0, b[n, t - 1] + 2*Sum[b[n - j^2, t - 1], {j, 1, Floor@Sqrt[n]}]]];
a[n_] := b[n^2, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 28 2023, after Alois P. Heinz *)

Prog

(PARI) {a(n)=local(THETA3=1+2*sum(m=1, n+1, x^(m^2))+x*O(x^(n^2))); polcoeff(THETA3^n, n^2)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A066535.

Main diagonal of A302996.

Sequence in context: A241589 A289776 A290169 * A067195 A080230 A084914

Adjacent sequences: A232170 A232171 A232172 * A232174 A232175 A232176

Keyword

nonn

Author

Paul D. Hanna, Nov 19 2013

Status

approved