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A232173 Number of ways of writing n^2 as a sum of n squares. 13
1, 2, 4, 30, 24, 1210, 18396, 235998, 4793456, 76168850, 1282320348, 25100418046, 481341997032, 10452086347274, 237925595533164, 5524220670435982, 136705837928870368, 3444192369181374754, 89772662325079950436, 2431910317560215089758, 67517711482300160612104 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..100

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.

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.

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

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Last modified February 22 05:17 EST 2019. Contains 320385 sequences. (Running on oeis4.)