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A201368 E.g.f.: 4*exp(4*x) / (5 - exp(4*x)).

%I A201368 #43 Feb 25 2024 12:43:41
%S A201368 1,5,30,230,2280,28280,421680,7336880,145879680,3263031680,
%T A201368 81097294080,2217097729280,66122900014080,2136392343342080,
%U A201368 74335250629908480,2771225281718343680,110198981079416340480,4655992415884353044480,208291013498682750074880,9835804726301090178990080
%N A201368 E.g.f.: 4*exp(4*x) / (5 - exp(4*x)).
%F A201368 O.g.f.: A(x) = Sum_{n>=0} n! * 5^n*x^n / Product_{k=0..n} (1+4*k*x).
%F A201368 O.g.f.: A(x) = 1/(1 - 5*x/(1-x/(1 - 10*x/(1-2*x/(1 - 15*x/(1-3*x/(1 - 20*x/(1-4*x/(1 - 25*x/(1-5*x/(1 - ...))))))))))), a continued fraction.
%F A201368 a(n) = Sum_{k=0..n} (-4)^(n-k) * 5^k * Stirling2(n,k) * k!.
%F A201368 a(n) = Sum_{k=0..n} A123125(n,k)*5^k. - _Philippe Deléham_, Nov 30 2011
%F A201368 a(n) ~ n! * (4/log(5))^(n+1). - _Vaclav Kotesovec_, Jun 13 2013
%F A201368 a(n) = 4^n*log(5) * Integral_{x = 0..oo} (ceiling(x))^n * 5^(-x) dx. - _Peter Bala_, Feb 06 2015
%F A201368 a(n) = 4^(n+1) * Sum_{k>=1} k^n / 5^k. - _Ilya Gutkovskiy_, Jun 28 2020
%e A201368 E.g.f.: E(x) = 1 + 5*x + 30*x^2/2! + 230*x^3/3! + 2280*x^4/4! + 28280*x^5/5! + ...
%e A201368 O.g.f.: A(x) = 1 + 5*x + 30*x^2 + 230*x^3 + 2280*x^4 + 28280*x^5 + ...
%e A201368 where A(x) = 1 + 5*x/(1+4*x) + 2!*5^2*x^2/((1+4*x)*(1+8*x)) + 3!*5^3*x^3/((1+4*x)*(1+8*x)*(1+12*x)) + 4!*5^4*x^4/((1+4*x)*(1+8*x)*(1+12*x)*(1+16*x)) + ...
%t A201368 Table[Sum[(-4)^(n-k)*5^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 13 2013 *)
%t A201368 With[{nn=20},CoefficientList[Series[4 Exp[4x]/(5-Exp[4x]),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Feb 25 2024 *)
%o A201368 (PARI) {a(n)=n!*polcoeff(4*exp(4*x+x*O(x^n))/(5 - exp(4*x+x*O(x^n))), n)}
%o A201368 (PARI) {a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+4*k*x+x*O(x^n))), n)}
%o A201368 (PARI) {a(n)=sum(k=0, n, (-4)^(n-k)*5^k*stirling(n, k, 2)*k!)}
%o A201368 (PARI) my(x='x+O('x^66)); Vec(serlaplace(4*exp(4*x)/(5-exp(4*x)))) \\ _Joerg Arndt_, May 06 2013
%o A201368 (Sage)
%o A201368 @CachedFunction
%o A201368 def BB(n, k, x):  # modified cardinal B-splines
%o A201368     if n == 1: return 0 if (x < 0) or (x >= k) else 1
%o A201368     return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)
%o A201368 def EulerianPolynomial(n, k, x):
%o A201368     if n == 0: return 1
%o A201368     return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))
%o A201368 [5^n*EulerianPolynomial(n, 1, 1/5) for n in (0..19)]   # _Peter Luschny_, May 04 2013
%Y A201368 Cf. A201365, A201366, A201367.
%K A201368 nonn,easy
%O A201368 0,2
%A A201368 _Paul D. Hanna_, Nov 30 2011