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A201355 Expansion of e.g.f.: 3*exp(3*x) / (4 - exp(3*x)).

%I A201355 #49 May 13 2025 20:42:33
%S A201355 1,4,20,132,1140,12324,160020,2424132,41967540,817374564,17688328020,
%T A201355 421061260932,10934334077940,307610736087204,9319558144624020,
%U A201355 302518807147502532,10474617188712332340,385347795973248950244,15010362565222418008020,617178205591321673884932
%N A201355 Expansion of e.g.f.: 3*exp(3*x) / (4 - exp(3*x)).
%H A201355 G. C. Greubel, <a href="/A201355/b201355.txt">Table of n, a(n) for n = 0..390</a>
%F A201355 O.g.f.: A(x) = Sum_{n>=0} n! * 4^n*x^n / Product_{k=0..n} (1+3*k*x).
%F A201355 O.g.f.: A(x) = 1/(1 - 4*x/(1-x/(1 - 8*x/(1-2*x/(1 - 12*x/(1-3*x/(1 - 16*x/(1-4*x/(1 - 20*x/(1-5*x/(1 - ...))))))))))), a continued fraction.
%F A201355 a(n) = Sum_{k=0..n} (-3)^(n-k) * 4^k * Stirling2(n,k) * k!.
%F A201355 a(n) = Sum_{k=0..n} A123125(n,k)*4^k. - _Philippe Deléham_, Nov 30 2011
%F A201355 G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 8*x*(k+1)/(8*x*(k+1) - 1 + 2*x*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 30 2013
%F A201355 a(n) ~ n! * (3/log(4))^(n+1) . - _Vaclav Kotesovec_, Jun 13 2013
%F A201355 a(n) = 3^n*log(4)*Integral_{x = 0..oo} (ceiling(x))^n * 4^(-x) dx. - _Peter Bala_, Feb 06 2015
%F A201355 a(n) = 3^(n+1) * Sum_{k>=1} k^n / 4^k. - _Ilya Gutkovskiy_, Jun 28 2020
%e A201355 E.g.f.: E(x) = 1 + 4*x + 20*x^2/2! + 132*x^3/3! + 1140*x^4/4! + 12324*x^5/5! + ...
%e A201355 O.g.f.: A(x) = 1 + 4*x + 20*x^2 + 132*x^3 + 1140*x^4 + 12324*x^5 + ...
%e A201355 where A(x) = 1 + 4*x/(1+3*x) + 2!*4^2*x^2/((1+3*x)*(1+6*x)) + 3!*4^3*x^3/((1+3*x)*(1+6*x)*(1+9*x)) + 4!*4^4*x^4/((1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)) + ...
%t A201355 Table[Sum[(-3)^(n-k)*4^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 13 2013 *)
%t A201355 With[{nn=20},CoefficientList[Series[3 Exp[3x]/(4-Exp[3x]),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 03 2020 *)
%o A201355 (PARI) {a(n)=n!*polcoeff(3*exp(3*x+x*O(x^n))/(4 - exp(3*x+x*O(x^n))), n)}
%o A201355 (PARI) {a(n)=polcoeff(sum(m=0, n, 4^m*m!*x^m/prod(k=1, m, 1+3*k*x+x*O(x^n))), n)}
%o A201355 (PARI) a(n)=sum(k=0, n, (-3)^(n-k)*4^k*stirling(n,k,2)*k!);
%o A201355 (PARI) my(x='x+O('x^66)); Vec(serlaplace(3*exp(3*x)/(4-exp(3*x)))) \\ _Joerg Arndt_, May 06 2013
%o A201355 (Sage)
%o A201355 @CachedFunction
%o A201355 def BB(n, k, x):  # modified cardinal B-splines
%o A201355     if n == 1: return 0 if (x < 0) or (x >= k) else 1
%o A201355     return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)
%o A201355 def EulerianPolynomial(n, k, x):
%o A201355     if n == 0: return 1
%o A201355     return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))
%o A201355 [4^n*EulerianPolynomial(n,1,1/4) for n in (0..19)]  # _Peter Luschny_, May 04 2013
%o A201355 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( 3*Exp(3*x)/(4-Exp(3*x)) ))); // _G. C. Greubel_, Jun 09 2022
%Y A201355 Cf. A201354, A000629, A123227.
%K A201355 nonn,easy
%O A201355 0,2
%A A201355 _Paul D. Hanna_, Nov 30 2011