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 A349088 #9 Apr 05 2022 12:29:40
%S A349088 0,1,2,6,25,125,750,5251,42008,378072,3780721,41587931,499055172,
%T A349088 6487717237,90828041318,1362420619770,21798729916321,370578408577457,
%U A349088 6670411354394226,126737815733490295,2534756314669805900,53229882608065923900,1171057417377450325801
%N A349088 a(n) = n! * Sum_{k=0..floor((n-1)/3)} 1 / (3*k+1)!.
%F A349088 E.g.f.: (exp(x) - 2 * exp(-x/2) * sin((Pi - 3*sqrt(3)*x)/6)) / (3*(1 - x)).
%F A349088 a(n) = floor(c * n!) for n > 0, where c = 1.041865355... = A143820.
%t A349088 Table[n! Sum[1/(3 k + 1)!, {k, 0, Floor[(n - 1)/3]}], {n, 0, 22}]
%t A349088 nmax = 22; CoefficientList[Series[(Exp[x] - 2 Exp[-x/2] Sin[(Pi - 3 Sqrt[3] x)/6])/(3 (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
%Y A349088 Cf. A002627, A009628, A143820, A186763, A337725, A349089, A352659.
%K A349088 nonn
%O A349088 0,3
%A A349088 _Ilya Gutkovskiy_, Mar 25 2022