A380275 Sum of the fourth powers of the coefficients of q in the q-factorials.

1, 1, 2, 34, 2710, 669142, 403186412, 504370709488, 1170803949124848, 4644277674894466168, 29557755573424568318844, 287158619888775996039794756, 4090368591132420991019182924018, 82628355729998755756059701468470738, 2301817961412922763844330401786521588244

Offset: 0

Author

Vaclav Kotesovec, Jan 18 2025

Keywords

• nonn

Comments

Conjecture: In general, sum of the k-th powers of the coefficients of q in the q-factorials is asymptotic to 2^((k-1)/2) * 3^(k-1) * n!^k / (sqrt(k) * Pi^((k-1)/2) * n^(3*(k-1)/2)).

Examples

			a(4) = 1^4 + 3^4 + 5^4 + 6^4 + 5^4 + 3^4 + 1^4 = 2710.
		

Links

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

Crossrefs

Cf. A008302, A127728, A380274.

Programs

Mathematica

Table[Total[CoefficientList[Expand[Product[Sum[x^i, {i, 0, m}], {m, 1, n-1}]], x]^4], {n, 0, 15}]

PARI

a(n) = my(v=Vec(prod(k=1, n, (1-q^k)/(1-q)))); sum(i=1, #v, v[i]^4); \\ Michel Marcus, Jan 18 2025

Formula

a(n) = Sum_{k>=0} A008302(n,k)^4.

Conjecture: a(n) ~ 27*sqrt(2) * n!^4 / (Pi^(3/2) * n^(9/2)).