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 A306082 #15 Feb 24 2025 08:52:27
%S A306082 1,1,3,13,99,901,8763,92653,1125939,16333141,274594923,5041348093,
%T A306082 97841114979,2007694705381,44043941312283,1036207737976333,
%U A306082 25969433606691219,688418684249653621,19275116061819888843,571069469474068377373,17898523203378840958659
%N A306082 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^(k^2)).
%C A306082 Conjecture: for positive integer k, reducing the sequence modulo k produces an eventually periodic sequence with period dividing phi(k) = A000010(k). For example, modulo 7 we obtain the sequence [1, 1, 3, 6, 1, 5, 6, 1, 3, 6, 1, 5, 6, 1, 3, 6, 1, 5, 6, ...], with an apparent period of 6 beginning at a(1). - _Peter Bala_, Feb 22 2025
%H A306082 Vaclav Kotesovec, <a href="/A306082/b306082.txt">Table of n, a(n) for n = 0..420</a>
%F A306082 a(n) = Sum_{k=0..n} Stirling2(n,k) * A001156(k) * k!.
%F A306082 a(n) ~ n! * exp(3 * 2^(-5/3) * Zeta(3/2)^(2/3) * (Pi*n/log(2))^(1/3)) * Zeta(3/2)^(2/3) / (2^(13/6) * sqrt(3) * Pi^(7/6) * n^(7/6) * (log(2))^(n - 1/6)).
%p A306082 a:=series(mul(1/(1-(exp(x)-1)^(k^2)),k=1..100),x=0,21): seq(n!*coeff(a, x, n), n=0..20); # _Paolo P. Lava_, Mar 26 2019
%t A306082 nmax = 20; CoefficientList[Series[Product[1/(1 - (Exp[x] - 1)^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
%Y A306082 Cf. A001156, A294529, A306083, A306147.
%K A306082 nonn
%O A306082 0,3
%A A306082 _Vaclav Kotesovec_, Jun 20 2018