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

1, 3, 12, 66, 480, 4368, 47712, 608016, 8855040, 145083648, 2641216512, 52891055616, 1155444326400, 27344999497728, 696933753434112, 19031293222127616, 554336947975618560, 17155693983744196608, 562168282464340672512, 19444889661250162262016

Offset: 0

Author

Philippe Deléham, Oct 06 2006

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..200
• Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.
• Eric Weisstein's MathWorld, Polylogarithm.
• Eric Weisstein's MathWorld, Lerch Transcendent.

Crossrefs

Cf. A000629, A201339, A122704, A009362, A123125, A028246.

Programs

Maple

a := n -> 2^(n+1)*polylog(-n, 1/3):
seq(round(evalf(a(n),32)), n=0..19); # Peter Luschny, Nov 03 2015
seq(expand(2^(n+1)*polylog(-n,1/3)), n=0..100); # Robert Israel, Nov 03 2015

Mathematica

CoefficientList[Series[2*Exp[2*x]/(3-Exp[2*x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 24 2013 *)
Round@Table[(-1)^(n+1) (LerchPhi[Sqrt[3], -n, 0] + LerchPhi[-Sqrt[3], -n, 0]), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

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 */

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 */

PARI

{Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0, n, (-2)^(n-k)*3^k*Stirling2(n, k)*k!)} /* Paul D. Hanna */

PARI

my(x='x+O('x^20)); Vec(serlaplace(2*exp(2*x)/(3-exp(2*x)))) \\ Joerg Arndt, May 06 2013

Sage

@CachedFunction
def BB(n, k, x):  # Modified Cardinal B-splines
    if n == 1: return 0 if (x < 0) or (x >= k) else 1
    return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)
def EulerianPolynomial(n, k, x):
    if n == 0: return 1
    return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))
def A123227(n) : return 3^n*EulerianPolynomial(n, 1, 1/3)
[A123227(n) for n in (0..18)]  # Peter Luschny, May 04 2013

Formula

a(n) = abs(A009362(n+1)).

a(n-1) = Sum_{k=1..n} 2^(n-k)*A028246(n,k), n>=1.

a(n) = Sum_{k=0..n} 3^k*A123125(n,k).

From Paul D. Hanna, Nov 30 2011: (Start)

a(n) = 3*A122704(n) for n>0.

a(n) = Sum_{k=0..n} (-2)^(n-k) * 3^k * Stirling2(n,k) * k!.

O.g.f.: Sum_{n>=0} 3^n * n!*x^n / Product_{k=0..n} (1+2*k*x).

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.

(End)

a(n) ~ n! * (2/log(3))^(n+1). - Vaclav Kotesovec, Jun 24 2013

a(n) = 2^n*log(3)*Integral_{x = 0..oo} (ceiling(x))^n * 3^(-x) dx. - Peter Bala, Feb 06 2015

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

a(n) = 2^(n+1)*Li_{-n}(1/3). - Peter Luschny, Nov 03 2015

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

Extensions

Name changed and a(8) corrected by Paul D. Hanna, Nov 30 2011