A380274 Sum of cubes of coefficients of q in the q-factorials.

1, 1, 2, 18, 522, 34986, 4524240, 1003172616, 351349509504, 182985164256000, 135303274820730372, 136936922140937021688, 184146557651652262521738, 321051865325352021467189658, 710866983641078174204266934736, 1964068265459581480020247325821224

Offset

0,3

Links

Table of n, a(n) for n=0..15.

Eric Weisstein's World of Mathematics, q-Factorial.

Formula

a(n) = Sum_{k>=0} A008302(n,k)^3.

Conjecture: a(n) ~ 6*sqrt(3) * n!^3 / (Pi * n^3).

Example

a(4) = 1^3 + 3^3 + 5^3 + 6^3 + 5^3 + 3^3 + 1^3 = 522.

Mathematica

Table[Total[CoefficientList[Expand[Product[Sum[x^i, {i, 0, m}], {m, 1, n-1}]], x]^3], {n, 0, 15}]

Prog

(PARI) a(n) = my(v=Vec(prod(k=1, n, (1-q^k)/(1-q)))); sum(i=1, #v, v[i]^3); \\ Michel Marcus, Jan 18 2025

Crossrefs

Cf. A008302, A127728, A380275.

Sequence in context: A208885 A255433 A121936 * A063389 A188202 A324308

Adjacent sequences: A380271 A380272 A380273 * A380275 A380277 A380278

Keyword

nonn

Author

Vaclav Kotesovec, Jan 18 2025

Status

approved