A347802 - OEIS

A347802

Expansion of ( Sum_{k>=0} k^2 * q^(k^2) )^3.

0, 0, 0, 1, 0, 0, 12, 0, 0, 48, 0, 27, 64, 0, 216, 0, 0, 432, 48, 243, 0, 384, 972, 0, 768, 0, 864, 804, 0, 3456, 600, 0, 0, 1968, 3888, 1350, 3072, 0, 5508, 0, 0, 7776, 2400, 6075, 1728, 9600, 1944, 0, 4096, 7776, 21600, 2022, 0, 3456, 17424, 0, 13824, 21552, 0, 19521, 0, 31104, 15984, 0, 0, 21600, 34896, 11907

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OFFSET

0,7

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Simon Plouffe, Numbers in the base e^Pi, arXiv:2509.15609 [math.NT], 2025. See p. 22/24, marked 215.

FORMULA

a(n) is sum of i^2 * j^2 * k^2 for positive integers i,j,k such that i^2+j^2+k^2=n.

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/512) / Pi^(9/4) / Gamma(3/4)^3 = A389058. - Simon Plouffe, Sep 22 2025

PROG

(PARI) a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, (i^2+j^2+k^2==n)*(i*j*k)^2)));

(PARI) my(N=66, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=0, sqrtint(N), k^2*x^k^2)^3))

CROSSREFS

Cf. A000408, A002102, A037215, A347801, A347803.

Sequence in context: A072837 A023917 A359001 * A230926 A064141 A230526

Adjacent sequences: A347799 A347800 A347801 * A347803 A347804 A347805

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Sep 14 2021

STATUS

approved