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Expansion of (1/(1-2x))*exp(2x/(1-2x)).
1

%I #11 Jan 19 2020 09:28:07

%S 1,4,28,272,3344,49472,852928,16758016,369082624,8996922368,

%T 240294124544,6974172532736,218457925292032,7342688736329728,

%U 263513011737051136,10054112734940561408,406301504472849907712,17333090863574658842624,778298003006760943353856

%N Expansion of (1/(1-2x))*exp(2x/(1-2x)).

%C a(n) = 2^n*A002720(n).

%F E.g.f.: (1/(1-2x))*exp(2x/(1-2x)). - corrected by _Vaclav Kotesovec_, Sep 26 2013

%F a(n) = 2^n*n!*sum{k=0..n, C(n,k)/k!}.

%F Conjecture: a(n) -4*n*a(n-1) +4*(n-1)^2*a(n-2)=0. - _R. J. Mathar_, Nov 14 2011

%F a(n) ~ 2^(n-1/2)*n^(n+1/4)*exp(2*sqrt(n)-n-1/2) * (1 + 31/(48*sqrt(n))). - _Vaclav Kotesovec_, Sep 26 2013

%F a(n) = 2^n*n!*LaguerreL(n, -1). - _Peter Luschny_, Jan 19 2020

%p a := n -> 2^n*n!*LaguerreL(n, -1):

%p seq(simplify(a(n)), n=0..18); # _Peter Luschny_, Jan 19 2020

%t Table[2^n*n!*Sum[Binomial[n,k]/k!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Sep 26 2013 *)

%t CoefficientList[Series[(1/(1-2*x))*E^(2*x/(1-2*x)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 26 2013 *)

%K easy,nonn

%O 0,2

%A _Paul Barry_, Apr 28 2007

%E Error in definition corrected by _Vaclav Kotesovec_, Sep 26 2013