A082032 Expansion of e.g.f.: exp(2*x)/(1-2*x).

1, 4, 20, 128, 1040, 10432, 125248, 1753600, 28057856, 505041920, 10100839424, 222218469376, 5333243269120, 138664325005312, 3882601100165120, 116478033004986368, 3727297056159629312, 126728099909427527680, 4562211596739391258624, 173364040676096868352000, 6934561627043874735128576

Offset: 0

Author

Paul Barry, Apr 02 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A010844. a(n) = b such that Integral_{x=0..1} (2*x)^n*exp(-x) dx = c - b*exp(-1). - Francesco Daddi, Jul 31 2011

Links

• Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Crossrefs

Cf. A000522, A010844, A082033.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Exp[2x]/(1-2x),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 02 2021 *)

PARI

my(x='x + O('x^25)); Vec(serlaplace(exp(2*x)/(1-2*x))) \\ Michel Marcus, Jan 27 2019

Formula

E.g.f.: exp(2*x)/(1-2*x)

a(n) = 2^n*A000522(n). - Vladeta Jovovic, Oct 29 2003

a(n) = 2n*a(n)+2^n, n>0, a(0)=1. - Paul Barry, Aug 26 2004

a(n) +2*(-n-1)*a(n-1) +4*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012

G.f.: 1/Q(0), where Q(k)= 1 - 2*x - 2*x*(k+1)/(1-2*x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013

G.f.: 1/Q(0), where Q(k) = 1 - 4*x*(k+1) - 4*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013

a(n) = 2^n*hypergeometric_U(1,n+2,1). - Peter Luschny, Nov 26 2014

Extensions

More terms from Michel Marcus, Jan 27 2019