OFFSET
0,3
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)).
Wang proves the general fixed-real-power asymptotic for sums of powers of Mahonian coefficients, and gives a third-order asymptotic expansion for this sequence; see the link. - Xinjun Wang, May 28 2026
LINKS
Xinjun Wang, Third-Order Asymptotics for Power Sums of Mahonian Coefficients and OEIS Conjectures A380274-A380275, Zenodo, 2026.
Xinjun Wang, Fixed-Power Sums of Mahonian Coefficients, ResearchGate (2026).
Eric Weisstein's World of Mathematics, q-Factorial.
FORMULA
a(n) = Sum_{j>=0} A008302(n,j)^4.
Conjecture: a(n) ~ 27*sqrt(2) * n!^4 / (Pi^(3/2) * n^(9/2)).
From Xinjun Wang, Jun 01 2026: (Start)
This conjectured asymptotic formula is proved by Wang, who also gives the following third-order refinement.
a(n) = 27*sqrt(2)/Pi^(3/2) * n!^4/n^(9/2) * (1 - 1143/(400*n) + 149174913/(15680000*n^2) - 190551792429/(6272000000*n^3) + o(n^(-3))).
More generally, for fixed real r > 1, Sum_j A008302(n,j)^r ~ 2^((r-1)/2)*3^(r-1)*n!^r/(sqrt(r)*Pi^((r-1)/2)*n^(3*(r-1)/2)). (End)
EXAMPLE
a(4) = 1^4 + 3^4 + 5^4 + 6^4 + 5^4 + 3^4 + 1^4 = 2710.
MATHEMATICA
Table[Total[CoefficientList[Expand[Product[Sum[x^i, {i, 0, m}], {m, 1, n-1}]], x]^4], {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]^4); \\ Michel Marcus, Jan 18 2025
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Vaclav Kotesovec, Jan 18 2025
STATUS
approved