A201355 - OEIS

%I #46 Aug 06 2022 07:24:06

%S 1,4,20,132,1140,12324,160020,2424132,41967540,817374564,17688328020,

%T 421061260932,10934334077940,307610736087204,9319558144624020,

%U 302518807147502532,10474617188712332340,385347795973248950244,15010362565222418008020,617178205591321673884932

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

%H G. C. Greubel, <a href="/A201355/b201355.txt">Table of n, a(n) for n = 0..390</a>

%F O.g.f.: A(x) = Sum_{n>=0} n! * 4^n*x^n / Product_{k=0..n} (1+3*k*x).

%F 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 a(n) = Sum_{k=0..n} (-3)^(n-k) * 4^k * Stirling2(n,k) * k!.

%F a(n) = Sum_{k=0..n} A123125(n,k)*4^k. - _Philippe Deléham_, Nov 30 2011

%F 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 a(n) ~ n! * (3/log(4))^(n+1) . - _Vaclav Kotesovec_, Jun 13 2013

%F a(n) = 3^n*log(4)*Integral_{x = 0..oo} (ceiling(x))^n * 4^(-x) dx. - _Peter Bala_, Feb 06 2015

%F a(n) = 3^(n+1) * Sum_{k>=1} k^n / 4^k. - _Ilya Gutkovskiy_, Jun 28 2020

%e 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 O.g.f.: A(x) = 1 + 4*x + 20*x^2 + 132*x^3 + 1140*x^4 + 12324*x^5 + ...

%e 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 Table[Sum[(-3)^(n-k)*4^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 13 2013 *)

%t 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 (PARI) {a(n)=n!*polcoeff(3*exp(3*x+x*O(x^n))/(4 - exp(3*x+x*O(x^n))), n)}

%o (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 (PARI) a(n)=sum(k=0, n, (-3)^(n-k)*4^k*stirling(n,k,2)*k!);

%o (PARI) my(x='x+O('x^66)); Vec(serlaplace(3*exp(3*x)/(4-exp(3*x)))) \\ _Joerg Arndt_, May 06 2013

%o (Sage)

%o @CachedFunction

%o def BB(n, k, x):  # modified cardinal B-splines

%o     if n == 1: return 0 if (x < 0) or (x >= k) else 1

%o     return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)

%o def EulerianPolynomial(n, k, x):

%o     if n == 0: return 1

%o     return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))

%o [4^n*EulerianPolynomial(n,1,1/4) for n in (0..19)]  # _Peter Luschny_, May 04 2013

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( 3*Exp(3*x)/(4-Exp(3*x)) ))); // _G. C. Greubel_, Jun 09 2022

%Y Cf. A201354, A000629, A123227.

%K nonn,easy

%O 0,2

%A _Paul D. Hanna_, Nov 30 2011