This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380275 #11 Feb 16 2025 08:34:07
%S A380275 1,1,2,34,2710,669142,403186412,504370709488,1170803949124848,
%T A380275 4644277674894466168,29557755573424568318844,
%U A380275 287158619888775996039794756,4090368591132420991019182924018,82628355729998755756059701468470738,2301817961412922763844330401786521588244
%N A380275 Sum of the fourth powers of the coefficients of q in the q-factorials.
%C A380275 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)).
%H A380275 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.
%F A380275 a(n) = Sum_{k>=0} A008302(n,k)^4.
%F A380275 Conjecture: a(n) ~ 27*sqrt(2) * n!^4 / (Pi^(3/2) * n^(9/2)).
%e A380275 a(4) = 1^4 + 3^4 + 5^4 + 6^4 + 5^4 + 3^4 + 1^4 = 2710.
%t A380275 Table[Total[CoefficientList[Expand[Product[Sum[x^i, {i, 0, m}], {m, 1, n-1}]], x]^4], {n, 0, 15}]
%o A380275 (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
%Y A380275 Cf. A008302, A127728, A380274.
%K A380275 nonn
%O A380275 0,3
%A A380275 _Vaclav Kotesovec_, Jan 18 2025