OFFSET
0,3
COMMENTS
Also the number of nonnegative integer solutions (a_1, a_2, ..., a_n) to the equation 1^2*a_1 + 2^2*a_2 + ... + n^2*a_n = n^3.
Also the number of partitions of n^3 into square parts not greater than n^2. - Paul D. Hanna, Feb 02 2024
EXAMPLE
1^2* 0 + 2^2*0 + 3^2*3 = 27.
1^2* 1 + 2^2*2 + 3^2*2 = 27.
1^2* 2 + 2^2*4 + 3^2*1 = 27.
1^2* 3 + 2^2*6 + 3^2*0 = 27.
1^2* 5 + 2^2*1 + 3^2*2 = 27.
1^2* 6 + 2^2*3 + 3^2*1 = 27.
1^2* 7 + 2^2*5 + 3^2*0 = 27.
1^2* 9 + 2^2*0 + 3^2*2 = 27.
1^2*10 + 2^2*2 + 3^2*1 = 27.
1^2*11 + 2^2*4 + 3^2*0 = 27.
1^2*14 + 2^2*1 + 3^2*1 = 27.
1^2*15 + 2^2*3 + 3^2*0 = 27.
1^2*18 + 2^2*0 + 3^2*1 = 27.
1^2*19 + 2^2*2 + 3^2*0 = 27.
1^2*23 + 2^2*1 + 3^2*0 = 27.
1^2*27 + 2^2*0 + 3^2*0 = 27.
So a(3) = 16.
PROG
(PARI) {a(n) = polcoeff(prod(i=1, n, sum(j=0, n^3\i^2, x^(i^2*j)+x*O(x^(n^3)))), n^3)}
(PARI) {a(n) = polcoeff( 1/prod(k=1, n, 1 - x^(k^2) +x*O(x^(n^3)) ), n^3) }
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 02 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 01 2018
STATUS
approved