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 A384522 #13 Jun 21 2025 13:59:27
%S A384522 1,3,30,426,8040,189768,5375280,177632976,6708685440,285038686848,
%T A384522 13456362881280,698786099602176,39586707755811840,2429498408440009728,
%U A384522 160571526535426529280,11370607719608891467776,858870213271187908362240,68928740686010010238353408
%N A384522 Expansion of e.g.f. 4/(7 - 3*exp(4*x)).
%H A384522 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polylogarithm">Polylogarithm</a>.
%F A384522 a(n) = (-4)^(n+1)/7 * Li_{-n}(7/3), where Li_{n}(x) is the polylogarithm function.
%F A384522 a(n) = 4^(n+1)/7 * Sum_{k>=0} k^n * (3/7)^k.
%F A384522 a(n) = Sum_{k=0..n} 3^k * 4^(n-k) * k! * Stirling2(n,k).
%F A384522 a(n) = (3/7) * Sum_{k=0..n} 7^k * (-4)^(n-k) * k! * Stirling2(n,k) for n > 0.
%F A384522 a(0) = 1; a(n) = 3 * Sum_{k=1..n} 4^(k-1) * binomial(n,k) * a(n-k).
%F A384522 a(0) = 1; a(n) = 3 * a(n-1) + 7 * Sum_{k=1..n-1} (-4)^(k-1) * binomial(n-1,k) * a(n-k).
%t A384522 With[{nn=20},CoefficientList[Series[4/(7-3Exp[4x]),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jun 21 2025 *)
%o A384522 (PARI) a(n) = (-4)^(n+1)*polylog(-n, 7/3)/7;
%Y A384522 Cf. A094419, A384514, A384521, A384523, A384524.
%K A384522 nonn
%O A384522 0,2
%A A384522 _Seiichi Manyama_, Jun 01 2025