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 A347989 #25 May 14 2025 04:36:29
%S A347989 1,5,71,1665,54649,2310945,119753843,7353403057,522289211873,
%T A347989 42137920501677,3807384320667135,380929847762489025,
%U A347989 41811136672902061321,4995760464106519955705,645541681316043216096315,89705032647088734873129825,13340173206548155385625683265,2114001534402053456524492822485
%N A347989 a(n) = [x^n] (2*n)! * Sum_{k=0..2*n} binomial(x+k,k).
%H A347989 Seiichi Manyama, <a href="/A347989/b347989.txt">Table of n, a(n) for n = 0..325</a>
%F A347989 a(n) = (2*n)! * Sum_{k=n..2*n} (2*n+1-k) * |Stirling1(k,n)|/k!.
%F A347989 a(n) = [x^(2*n)] ((2*n)!/n!) * (-log(1 - x))^n/(1 - x)^2.
%F A347989 From _Vaclav Kotesovec_, Sep 23 2021, updated May 14 2025: (Start)
%F A347989 a(n) = [x^n] Gamma(2*n + x + 2) / Gamma(x + 2).
%F A347989 a(n) ~ c * d^n * (n-1)!, where d = 8*w^2/(2*w-1), w = -LambertW(-1,-exp(-1/2)/2) and c = w^2 * sqrt(2) / (sqrt(w-1)*Pi) = 1.5967712192197964362930380385801737624829174112909160160618... (End)
%o A347989 (PARI) a(n) = (2*n)!*polcoef(sum(k=n, 2*n, binomial(x+k, k)), n);
%o A347989 (PARI) a(n) = (2*n)!*sum(k=n, 2*n, (2*n+1-k)*abs(stirling(k, n, 1))/k!);
%Y A347989 Cf. A001706, A001707, A001708, A001709, A008275, A143491, A347987.
%K A347989 nonn
%O A347989 0,2
%A A347989 _Seiichi Manyama_, Sep 23 2021