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A052838 Expansion of e.g.f.: (exp(x/(1-x)) - 1)^2.

%I A052838 #20 Feb 24 2021 08:19:46
%S A052838 0,0,2,18,158,1510,15962,186270,2385182,33290862,503277242,8193803926,
%T A052838 142938943886,2659770747270,52581058479770,1100423513438766,
%U A052838 24302677755662654,564770268904566238
%N A052838 Expansion of e.g.f.: (exp(x/(1-x)) - 1)^2.
%C A052838 Previous name was: A simple grammar.
%H A052838 G. C. Greubel, <a href="/A052838/b052838.txt">Table of n, a(n) for n = 0..250</a>
%H A052838 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=805">Encyclopedia of Combinatorial Structures 805</a>
%H A052838 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A052838 E.g.f.: (exp(-x/(-1+x)) - 1)^2.
%F A052838 Recurrence: n*(2 +5*n +4*n^2 +n^3)*a(n) - (18 +35*n +21*n^2 +4*n^3)*a(n+1) +2*(19 +15*n +3*n^2)*a(n+2) - (13 +4*n)*a(n+3) + a(n+4) = 0, with a(1)=0, a(0)=0, a(2)=2, a(3)=18.
%F A052838 From _Vaclav Kotesovec_, Sep 30 2013: (Start)
%F A052838 a(n) = A052897(n) - 2*A000262(n) for n > 0.
%F A052838 a(n) ~ 2^(-1/4)*exp(2*sqrt(2*n)-n-1)*n^(n-1/4). (End)
%F A052838 From _G. C. Greubel_, Feb 23 2021: (Start)
%F A052838 a(n) = 2 * n! * Sum_{j=0..n-1} binomial(n-1, j)*(2^j -1)/(j+1)!.
%F A052838 a(n) = n! * (LaguerreL(n, -1, -2) - 2*LaguerreL(n, -1, -1)) + [n=0]. (End)
%p A052838 spec := [S,{C=Sequence(Z,1 <= card),B=Set(C,1 <= card),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052838 CoefficientList[Series[(E^(x/(1-x))-1)^2, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)
%t A052838 Table[n!*(LaguerreL[n, -1, -2] - 2* LaguerreL[n, -1, -1]) + Boole[n==0], {n,0,20}] (* _G. C. Greubel_, Feb 23 2021 *)
%o A052838 (Sage) [0]+[factorial(n)*(gen_laguerre(n, -1, -2) - 2*gen_laguerre(n, -1, -1)) for n in (1..25)] # _G. C. Greubel_, Feb 23 2021
%o A052838 (Magma) [0] cat [2*Factorial(n)*(&+[Binomial(n-1,j)*(2^j-1)/Factorial(j+1): j in [0..n-1]]) : n in [1..25]]; // _G. C. Greubel_, Feb 23 2021
%Y A052838 Cf. A000262, A052897.
%K A052838 easy,nonn
%O A052838 0,3
%A A052838 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052838 New name, using e.g.f., by _Vaclav Kotesovec_, Sep 30 2013