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 A355171 #24 Feb 17 2024 04:50:21
%S A355171 0,1,7,50,406,3804,41028,506064,7084656,111690720,1967421600,
%T A355171 38425449600,825970435200,19404363283200,495012834489600,
%U A355171 13632039812966400,403120633444300800,12740557701389414400,428546132879432601600,15284163618598275072000,576073025410937628672000
%N A355171 a(n) = Sum_{k=0..n} binomial(n, k + 1)*k!*(n + 1)!/(k + 2)!.
%F A355171 a(n) = n*(n + 1)!*hypergeom([1, 1, 1 - n], [2, 3], -1) / 2.
%F A355171 a(n) = Sum_{k=0..n} (-1)^(k+1)*k!*A066667(n, k + 1).
%F A355171 E.g.f.: log((1 - x) / (1 - 2*x)) / (1 - x)^2. - _Mélika Tebni_, Jun 23 2022
%F A355171 a(n) ~ 2^(n+2) * (n-1)!. - _Vaclav Kotesovec_, Feb 17 2024
%p A355171 a := n -> n*(n + 1)!*hypergeom([1, 1, 1 - n], [2, 3], -1) / 2;
%p A355171 seq(simplify(a(n)), n = 0..20);
%t A355171 a[n_] := n * (n + 1)! * HypergeometricPFQ[{1, 1, 1 - n}, {2, 3}, -1]/2; Array[a, 21, 0] (* _Amiram Eldar_, Jun 22 2022 *)
%o A355171 (Python)
%o A355171 from math import comb, factorial
%o A355171 def A355171(n):
%o A355171 f = factorial(n+1)
%o A355171 return sum(f*comb(n,k+1)//(k+2)//(k+1) for k in range(n+1)) # _Chai Wah Wu_, Jun 22 2022
%Y A355171 Cf. A066667, A001705, A103213.
%K A355171 nonn
%O A355171 0,3
%A A355171 _Peter Luschny_, Jun 22 2022