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
Keywords
Examples
a(4) = 1^3 + 3^3 + 5^3 + 6^3 + 5^3 + 3^3 + 1^3 = 522.
Links
- Eric Weisstein's World of Mathematics, q-Factorial.
Programs
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Mathematica
Table[Total[CoefficientList[Expand[Product[Sum[x^i, {i, 0, m}], {m, 1, n-1}]], x]^3], {n, 0, 15}] -
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
Formula
a(n) = Sum_{k>=0} A008302(n,k)^3.
Conjecture: a(n) ~ 6*sqrt(3) * n!^3 / (Pi * n^3).