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%I #62 Sep 08 2022 08:44:59
%S 1,2,8,44,304,2512,24064,261536,3173888,42483968,621159424,9841950208,
%T 167879268352,3065723549696,59651093528576,1231571119812608,
%U 26883546193002496,618463501807058944
%N Expansion of e.g.f.: exp(2*x/(1-x)).
%C Previous name was: A simple grammar.
%H Vincenzo Librandi, <a href="/A052897/b052897.txt">Table of n, a(n) for n = 0..200</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=873">Encyclopedia of Combinatorial Structures 873</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F E.g.f.: exp(2*x/(1-x)). - _Vladeta Jovovic_, Jan 04 2001
%F Recurrence: {a(0)=1, a(1)=2, (n^2+n)*a(n) + (-4-2*n)*a(n+1) + a(n+2)}.
%F LAH transform of A000079: a(n) = Sum_{k=0..n) 2^k*n!/k!*binomial(n-1, k-1). - _Vladeta Jovovic_, Oct 17 2003
%F a(n) = n!*L(n,-1,-2). - _Karol A. Penson_, Oct 16 2006 [Here L(n,a,x) is the n-th generalized Laguerre polynomial with parameter a, evaluated at x. L(n,a,x) is 1 if n=0, a+1-x if n=1 and otherwise (2*n+a-1-x)/n*L(n-1,a,x)-(n+a-1)/n*L(n-2,a,x). - _Peter Luschny_, Nov 20 2011]
%F a(n) ~ 2^(-1/4)*exp(2*sqrt(2*n)-n-1)*n^(n-1/4) * (1 + 7/(48*sqrt(2*n))). - _Vaclav Kotesovec_, Oct 09 2012, extended Dec 01 2021
%F E.g.f.: 1 + 2*x/((1-x)*T(0) - x), where T(k) = 4*k+1 + x^2/((4*k+3)*(1-x)^2 + x^2/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 30 2013
%F E.g.f.: exp(Sum_{k>=1} 2*x^k). - _Vaclav Kotesovec_, Mar 07 2015
%F a(n) = Sum_{k=0..n} binomial(n,k)*l(k)*l(n-k), where l(m) = A000262(m). - _Emanuele Munarini_, Aug 31 2017
%p L := proc(n,a,x) if n=0 then 1 elif n=1 then a+1-x else (2*n+a-1-x)/n*L(n-1,a,x) - (n+a-1)/n*L(n-2,a,x) fi end: A052897 := n -> n!*L(n,-1,-2): seq(A052897(n),n=0..17); # _Peter Luschny_, Nov 20 2011
%p spec := [S,{B=Set(C),C=Sequence(Z,1 <= card),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t Range[0, 19]! CoefficientList[ Series[E^(2*x/(1 - x)), {x, 0, 19}], x] (* _Zerinvary Lajos_, Mar 21 2007 *)
%t Table[n!*LaguerreL[n, -1, -2], {n,0,30}] (* _G. C. Greubel_, Feb 23 2021 *)
%o (PARI) a=Vec(exp(2*x/(1-x)));for(n=2,#a-1,a[n+1]*=n!);a \\ _Charles R Greathouse IV_, Nov 20 2011
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(2*x/(1 - x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 15 2018
%o (Magma) [Factorial(n)*Evaluate(LaguerrePolynomial(n, -1), -2): n in [0..25]]; // _G. C. Greubel_, Feb 23 2021
%o (Sage) [factorial(n)*gen_laguerre(n, -1, -2) for n in (0..25)] # _G. C. Greubel_, Feb 23 2021
%Y Row sums of A059110.
%Y Cf. A000262, A025168, A255806, A317364.
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E New name using e.g.f., _Vaclav Kotesovec_, Feb 25 2014
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