A321238 a(n) = [x^(n^3)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).

1, 1, 1, 4, 16, 87, 911, 8081, 82494, 1108584, 14559487, 206462480, 3300362073, 54235076625, 939612600043, 17366394088532, 332129019947772, 6615538793829307, 137564490944940832, 2954281836759475893, 65572183746807351880, 1503752010271535590284, 35476544827929325305961

Offset

0,4

Comments

Also the number of nonnegative integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n^3.

Links

Table of n, a(n) for n=0..22.

Example

1*0^2 + 2*0^2 + 3*0^2 + 4*4^2 = 64.
1*0^2 + 2*0^2 + 3*4^2 + 4*2^2 = 64.
1*1^2 + 2*0^2 + 3*3^2 + 4*3^2 = 64.
1*1^2 + 2*4^2 + 3*3^2 + 4*1^2 = 64.
1*2^2 + 2*2^2 + 3*4^2 + 4*1^2 = 64.
1*2^2 + 2*4^2 + 3*2^2 + 4*2^2 = 64.
1*4^2 + 2*0^2 + 3*2^2 + 4*3^2 = 64.
1*4^2 + 2*0^2 + 3*4^2 + 4*0^2 = 64.
1*4^2 + 2*4^2 + 3*0^2 + 4*2^2 = 64.
1*4^2 + 2*4^2 + 3*2^2 + 4*1^2 = 64.
1*5^2 + 2*0^2 + 3*1^2 + 4*3^2 = 64.
1*5^2 + 2*2^2 + 3*3^2 + 4*1^2 = 64.
1*5^2 + 2*4^2 + 3*1^2 + 4*1^2 = 64.
1*6^2 + 2*0^2 + 3*2^2 + 4*2^2 = 64.
1*7^2 + 2*2^2 + 3*1^2 + 4*1^2 = 64.
1*8^2 + 2*0^2 + 3*0^2 + 4*0^2 = 64.
So a(4) = 16.

Prog

(PARI) {a(n) = polcoeff(prod(i=1, n, sum(j=0, sqrtint(n^3\i), x^(i*j^2)+x*O(x^(n^3)))), n^3)}

Crossrefs

Cf. A300446, A321139, A321239.

Sequence in context: A184507 A165964 A300279 * A005618 A005495 A052124

Adjacent sequences: A321235 A321236 A321237 * A321239 A321240 A321241

Keyword

nonn

Author

Seiichi Manyama, Nov 01 2018

Status

approved