A380274 - OEIS

%I #10 Jan 20 2025 03:27:39

%S 1,1,2,18,522,34986,4524240,1003172616,351349509504,182985164256000,

%T 135303274820730372,136936922140937021688,184146557651652262521738,

%U 321051865325352021467189658,710866983641078174204266934736,1964068265459581480020247325821224

%N Sum of cubes of coefficients of q in the q-factorials.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.

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

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

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

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

%o (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

%Y Cf. A008302, A127728, A380275.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Jan 18 2025