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 A201366 #27 Aug 06 2022 07:24:15
%S A201366 1,5,40,470,7360,144080,3384640,92761520,2905461760,102379969280,
%T A201366 4008411658240,172632406008320,8110747682652160,412820794294292480,
%U A201366 22628039202542755840,1328909797186015877120,83247808119808161218560,5540883903212529402183680,390489065613179063896637440
%N A201366 E.g.f.: 2*exp(2*x) / (5 - 3*exp(2*x)).
%F A201366 O.g.f.: A(x) = Sum_{n>=0} n! * 5^n*x^n / Product_{k=0..n} (1+2*k*x).
%F A201366 O.g.f.: A(x) = 1/(1 - 5*x/(1-3*x/(1 - 10*x/(1-6*x/(1 - 15*x/(1-9*x/(1 - 20*x/(1-12*x/(1 - 25*x/(1-15*x/(1 - ...))))))))))), a continued fraction.
%F A201366 a(n) = Sum_{k=0..n} (-2)^(n-k) * 5^k * Stirling2(n,k) * k!.
%F A201366 a(n) = Sum_{k=0..n} A123125(n,k)*5^k*3^(n-k). - _Philippe Deléham_, Nov 30 2011
%F A201366 a(n) ~ n! / (3*(log(5/3)/2)^(n+1)). - _Vaclav Kotesovec_, Jun 13 2013
%F A201366 a(n) = 2^n*log(5/3) * Integral_{x = 0..oo} (ceiling(x))^n * (5/3)^(-x) dx. - _Peter Bala_, Feb 06 2015
%e A201366 E.g.f.: E(x) = 1 + 5*x + 40*x^2/2! + 470*x^3/3! + 7360*x^4/4! + 144080*x^5/5! + ...
%e A201366 O.g.f.: A(x) = 1 + 5*x + 40*x^2 + 470*x^3 + 7360*x^4 + 144080*x^5 + ...
%e A201366 where A(x) = 1 + 5*x/(1+2*x) + 2!*5^2*x^2/((1+2*x)*(1+4*x)) + 3!*5^3*x^3/((1+2*x)*(1+4*x)*(1+6*x)) + 4!*5^4*x^4/((1+2*x)*(1+4*x)*(1+6*x)*(1+8*x)) + ...
%t A201366 Table[Sum[(-2)^(n-k)*5^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 13 2013 *)
%t A201366 With[{nn=20},CoefficientList[Series[(2Exp[2x])/(5-3Exp[2x]),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Dec 29 2018 *)
%o A201366 (PARI) {a(n)=n!*polcoeff(2*exp(2*x+x*O(x^n))/(5 - 3*exp(2*x+x*O(x^n))), n)}
%o A201366 (PARI) {a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+2*k*x+x*O(x^n))), n)}
%o A201366 (PARI) {Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
%o A201366 {a(n)=sum(k=0, n, (-2)^(n-k)*5^k*Stirling2(n, k)*k!)}
%Y A201366 Cf. A201365, A201367, A201368.
%K A201366 nonn,easy
%O A201366 0,2
%A A201366 _Paul D. Hanna_, Nov 30 2011