A329266 a(n) is the number of solutions of the infinite Diophantine equation Sum_{j>0} j^r*(k_j)^2 = n with k_j integers and r = 4.

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 8, 4, 4, 0, 0, 4, 0, 0

Offset

0,2

Links

Table of n, a(n) for n=0..87.

Nian Hong Zhou, Yalin Sun, Counting the number of solutions to certain infinite Diophantine equations, arXiv:1910.07884 [math.NT], 2019.

Formula

a(n) = [q^n] Product_{j>0} Product_{n>0} (1 - (-1)^n*q^(n*j^r)) / (1 + (-1)^n*q^(n*j^r)) with r = 4 (see Proposition 1.1 in Zhou and Sun).

Example

a(25) = 6 since there are 6 integer solutions to 1^4*k1^2 + 2^4*k2^2 + ... = 25:
k1 = +-5 and k_j = 0 for j > 1;
k1 = -3, k2 = +-1 and k_j = 0 for j > 2;
k1 = 3, k2 = +-1 and k_j = 0 for j > 2.

Mathematica

nmax=87; r=4; CoefficientList[Series[Product[Product[(1-(-1)^n*q^(n*j^r))/(1+(-1)^n*q^(n*j^r)), {n, 1, nmax}], {j, 1, nmax}], {q, 0, nmax}], q]

Crossrefs

Cf. A000041, A000122, A320067 (r = 1), A320068, A320078, A320968, A320992, A329264 (r = 2), A329265 (r = 3).

Sequence in context: A318984 A227395 A255258 * A260671 A033725 A204010

Adjacent sequences: A329263 A329264 A329265 * A329267 A329268 A329269

Keyword

nonn

Author

Stefano Spezia, Nov 09 2019

Status

approved