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

1, 4, 20, 132, 1140, 12324, 160020, 2424132, 41967540, 817374564, 17688328020, 421061260932, 10934334077940, 307610736087204, 9319558144624020, 302518807147502532, 10474617188712332340, 385347795973248950244, 15010362565222418008020, 617178205591321673884932

Offset: 0

Author

Paul D. Hanna, Nov 30 2011

Keywords

• nonn
• easy

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..390

Crossrefs

Cf. A201354, A000629, A123227.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( 3*Exp(3*x)/(4-Exp(3*x)) ))); // G. C. Greubel, Jun 09 2022

Mathematica

Table[Sum[(-3)^(n-k)*4^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 13 2013 *)
With[{nn=20},CoefficientList[Series[3 Exp[3x]/(4-Exp[3x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 03 2020 *)

PARI

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

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)}

PARI

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

PARI

my(x='x+O('x^66)); Vec(serlaplace(3*exp(3*x)/(4-exp(3*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))
[4^n*EulerianPolynomial(n,1,1/4) for n in (0..19)]  # Peter Luschny, May 04 2013

Formula

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

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.

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

a(n) = Sum_{k=0..n} A123125(n,k)*4^k. - Philippe Deléham, Nov 30 2011

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

a(n) ~ n! * (3/log(4))^(n+1) . - Vaclav Kotesovec, Jun 13 2013

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

a(n) = 3^(n+1) * Sum_{k>=1} k^n / 4^k. - Ilya Gutkovskiy, Jun 28 2020