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 A201367 #36 Sep 22 2024 11:31:45
%S A201367 1,5,35,345,4515,73905,1451835,33273945,871529715,25681042305,
%T A201367 840815302635,30281769805545,1189735610250915,50638609760802705,
%U A201367 2321120945531697435,113992686944812385145,5971520591679167948115,332369999588147180115105,19587647624733292373756235
%N A201367 E.g.f.: 3*exp(3*x) / (5 - 2*exp(3*x)).
%H A201367 Robert Israel, <a href="/A201367/b201367.txt">Table of n, a(n) for n = 0..250</a>
%F A201367 O.g.f.: A(x) = Sum_{n>=0} n! * 5^n*x^n / Product_{k=0..n} (1+3*k*x).
%F A201367 O.g.f.: A(x) = 1/(1 - 5*x/(1-2*x/(1 - 10*x/(1-4*x/(1 - 15*x/(1-6*x/(1 - 20*x/(1-8*x/(1 - 25*x/(1-10*x/(1 - ...))))))))))), a continued fraction.
%F A201367 a(n) = Sum_{k=0..n} (-3)^(n-k) * 5^k * Stirling2(n,k) * k!.
%F A201367 a(n) = Sum_{k=0..n} A123125(n,k)*5^k*2*(n-k). - _Philippe Deléham_, Nov 30 2011
%F A201367 a(n) ~ n! / (2*(log(5/2)/3)^(n+1)). - _Vaclav Kotesovec_, Jun 13 2013
%F A201367 a(n) = 3^n*log(5/2) * Integral_{x = 0..oo} (ceiling(x))^n * (5/2)^(-x) dx. - _Peter Bala_, Feb 06 2015
%e A201367 E.g.f.: E(x) = 1 + 5*x + 35*x^2/2! + 345*x^3/3! + 4515*x^4/4! + 73905*x^5/5! + ...
%e A201367 O.g.f.: A(x) = 1 + 5*x + 35*x^2 + 345*x^3 + 4515*x^4 + 73905*x^5 + ...
%e A201367 where A(x) = 1 + 5*x/(1+3*x) + 2!*5^2*x^2/((1+3*x)*(1+6*x)) + 3!*5^3*x^3/((1+3*x)*(1+6*x)*(1+9*x)) + 4!*5^4*x^4/((1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)) + ...
%p A201367 S:= series(3*exp(3*x)/(5-2*exp(3*x)),x,51):
%p A201367 seq(coeff(S,x,n)*n!,n=0..50); # _Robert Israel_, Nov 18 2019
%t A201367 Table[Sum[(-3)^(n-k)*5^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 13 2013 *)
%t A201367 With[{nn=20},CoefficientList[Series[(3*Exp[3x])/(5-2*Exp[3x]),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Sep 07 2024 *)
%o A201367 (PARI) {a(n)=n!*polcoeff(3*exp(3*x+x*O(x^n))/(5 - 2*exp(3*x+x*O(x^n))), n)}
%o A201367 (PARI) {a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+3*k*x+x*O(x^n))), n)}
%o A201367 (PARI) {Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
%o A201367 {a(n)=sum(k=0, n, (-3)^(n-k)*5^k*Stirling2(n, k)*k!)}
%Y A201367 Cf. A201365, A201366, A201368.
%K A201367 nonn,easy
%O A201367 0,2
%A A201367 _Paul D. Hanna_, Nov 30 2011