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A052897
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Expansion of e.g.f.: exp(2*x/(1-x)).
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11
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1, 2, 8, 44, 304, 2512, 24064, 261536, 3173888, 42483968, 621159424, 9841950208, 167879268352, 3065723549696, 59651093528576, 1231571119812608, 26883546193002496, 618463501807058944
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OFFSET
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0,2
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COMMENTS
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Previous name was: A simple grammar.
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LINKS
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FORMULA
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Recurrence: {a(0)=1, a(1)=2, (n^2+n)*a(n) + (-4-2*n)*a(n+1) + a(n+2)}.
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]
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
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
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MAPLE
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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
spec := [S, {B=Set(C), C=Sequence(Z, 1 <= card), S=Prod(B, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
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Range[0, 19]! CoefficientList[ Series[E^(2*x/(1 - x)), {x, 0, 19}], x] (* Zerinvary Lajos, Mar 21 2007 *)
Table[n!*LaguerreL[n, -1, -2], {n, 0, 30}] (* G. C. Greubel, Feb 23 2021 *)
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PROG
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(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
(Magma) [Factorial(n)*Evaluate(LaguerrePolynomial(n, -1), -2): n in [0..25]]; // G. C. Greubel, Feb 23 2021
(Sage) [factorial(n)*gen_laguerre(n, -1, -2) for n in (0..25)] # G. C. Greubel, Feb 23 2021
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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STATUS
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approved
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