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A123227 Expansion of e.g.f.: 2*exp(2*x) / (3 - exp(2*x)).

%I A123227 #59 Apr 07 2025 09:50:53
%S A123227 1,3,12,66,480,4368,47712,608016,8855040,145083648,2641216512,
%T A123227 52891055616,1155444326400,27344999497728,696933753434112,
%U A123227 19031293222127616,554336947975618560,17155693983744196608,562168282464340672512,19444889661250162262016
%N A123227 Expansion of e.g.f.: 2*exp(2*x) / (3 - exp(2*x)).
%H A123227 G. C. Greubel, <a href="/A123227/b123227.txt">Table of n, a(n) for n = 0..200</a>
%H A123227 Paul Barry, <a href="https://arxiv.org/abs/1702.04007">Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays</a>, arXiv:1702.04007 [math.CO], 2017.
%H A123227 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%H A123227 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/LerchTranscendent.html">Lerch Transcendent</a>.
%F A123227 a(n) = abs(A009362(n+1)).
%F A123227 a(n-1) = Sum_{k=1..n} 2^(n-k)*A028246(n,k), n>=1.
%F A123227 a(n) = Sum_{k=0..n} 3^k*A123125(n,k).
%F A123227 From _Paul D. Hanna_, Nov 30 2011: (Start)
%F A123227 a(n) = 3*A122704(n) for n>0.
%F A123227 a(n) = Sum_{k=0..n} (-2)^(n-k) * 3^k * Stirling2(n,k) * k!.
%F A123227 O.g.f.: Sum_{n>=0} 3^n * n!*x^n / Product_{k=0..n} (1+2*k*x).
%F A123227 O.g.f.: 1/(1 - 3*x/(1-x/(1 - 6*x/(1-2*x/(1 - 9*x/(1-3*x/(1 - 12*x/(1-4*x/(1 - 15*x/(1-5*x/(1 - ...))))))))))), a continued fraction.
%F A123227 (End)
%F A123227 a(n) ~ n! * (2/log(3))^(n+1). - _Vaclav Kotesovec_, Jun 24 2013
%F A123227 a(n) = 2^n*log(3)*Integral_{x = 0..oo} (ceiling(x))^n * 3^(-x) dx. - _Peter Bala_, Feb 06 2015
%F A123227 a(n) = (-1)^(n+1)*(LerchPhi(sqrt(3), -n, 0) + LerchPhi(-sqrt(3), -n, 0)) = (-1)^(n+1)*(Li_{-n}(sqrt(3)) + Li_{-n}(-sqrt(3))) - 2*0^n, where Li_n(x) is the polylogarithm. - _Vladimir Reshetnikov_, Oct 31 2015
%F A123227 a(n) = 2^(n+1)*Li_{-n}(1/3). - _Peter Luschny_, Nov 03 2015
%F A123227 a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1). - _Ilya Gutkovskiy_, Jul 05 2020
%p A123227 a := n -> 2^(n+1)*polylog(-n, 1/3):
%p A123227 seq(round(evalf(a(n),32)), n=0..19); # _Peter Luschny_, Nov 03 2015
%p A123227 seq(expand(2^(n+1)*polylog(-n,1/3)), n=0..100); # _Robert Israel_, Nov 03 2015
%t A123227 CoefficientList[Series[2*Exp[2*x]/(3-Exp[2*x]), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 24 2013 *)
%t A123227 Round@Table[(-1)^(n+1) (LerchPhi[Sqrt[3], -n, 0] + LerchPhi[-Sqrt[3], -n, 0]), {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 31 2015 *)
%o A123227 (PARI) {a(n)=n!*polcoeff(2*exp(2*x+x*O(x^n))/(3 - exp(2*x+x*O(x^n))), n)} /* _Paul D. Hanna_ */
%o A123227 (PARI) {a(n)=polcoeff(sum(m=0, n, 3^m*m!*x^m/prod(k=1, m, 1+2*k*x+x*O(x^n))), n)} /* _Paul D. Hanna_ */
%o A123227 (PARI) {Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
%o A123227 {a(n)=sum(k=0, n, (-2)^(n-k)*3^k*Stirling2(n, k)*k!)} /* _Paul D. Hanna_ */
%o A123227 (Sage)
%o A123227 @CachedFunction
%o A123227 def BB(n, k, x):  # Modified Cardinal B-splines
%o A123227     if n == 1: return 0 if (x < 0) or (x >= k) else 1
%o A123227     return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)
%o A123227 def EulerianPolynomial(n, k, x):
%o A123227     if n == 0: return 1
%o A123227     return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))
%o A123227 def A123227(n) : return 3^n*EulerianPolynomial(n, 1, 1/3)
%o A123227 [A123227(n) for n in (0..18)]  # _Peter Luschny_, May 04 2013
%o A123227 (PARI) my(x='x+O('x^20)); Vec(serlaplace(2*exp(2*x)/(3-exp(2*x)))) \\ _Joerg Arndt_, May 06 2013
%Y A123227 Cf. A000629, A201339, A122704, A009362, A123125, A028246.
%K A123227 nonn,easy
%O A123227 0,2
%A A123227 _Philippe Deléham_, Oct 06 2006
%E A123227 Name changed and a(8) corrected by _Paul D. Hanna_, Nov 30 2011