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%I #33 Aug 02 2021 15:11:37
%S 1,4,20,128,1040,10432,125248,1753600,28057856,505041920,10100839424,
%T 222218469376,5333243269120,138664325005312,3882601100165120,
%U 116478033004986368,3727297056159629312,126728099909427527680,4562211596739391258624,173364040676096868352000,6934561627043874735128576
%N Expansion of e.g.f.: exp(2*x)/(1-2*x).
%C 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
%H Michael Z. Spivey and Laura L. Steil, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%F E.g.f.: exp(2*x)/(1-2*x)
%F a(n) = 2^n*A000522(n). - _Vladeta Jovovic_, Oct 29 2003
%F a(n) = 2n*a(n)+2^n, n>0, a(0)=1. - _Paul Barry_, Aug 26 2004
%F a(n) +2*(-n-1)*a(n-1) +4*(n-1)*a(n-2)=0. - _R. J. Mathar_, Nov 26 2012
%F 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
%F 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
%F a(n) = 2^n*hypergeometric_U(1,n+2,1). - _Peter Luschny_, Nov 26 2014
%t With[{nn=30},CoefficientList[Series[Exp[2x]/(1-2x),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Aug 02 2021 *)
%o (PARI) my(x='x + O('x^25)); Vec(serlaplace(exp(2*x)/(1-2*x))) \\ _Michel Marcus_, Jan 27 2019
%Y Cf. A000522, A010844, A082033.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 02 2003
%E More terms from _Michel Marcus_, Jan 27 2019
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