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A simple grammar.
2

%I #34 Jun 03 2024 12:26:50

%S 0,1,2,15,184,3145,68976,1846999,58413440,2130740721,88061420800,

%T 4066862460991,207556068584448,11600364266112505,704664527894104064,

%U 46226086991634882375,3256882066245640093696,245279323467051422886241

%N A simple grammar.

%H G. C. Greubel, <a href="/A052857/b052857.txt">Table of n, a(n) for n = 0..250</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=825">Encyclopedia of Combinatorial Structures 825</a>

%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>

%F E.g.f.: exp(RootOf(exp(_Z)*x*_Z+exp(_Z)*x-_Z))*x.

%F a(n) = (n-1)!*Sum_{k=1..n-1} n^k*binomial(n-2,k-1)/k!, a(1)=1. - _Vladimir Kruchinin_, May 10 2011

%F a(n) = n!*hypergeom([-n+2], [2], -n) for n>=2. - _Peter Luschny_, Apr 20 2016

%F a(n) ~ exp(n/phi - n) * phi^(2*n-2) * n^(n-1) / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Oct 01 2017

%F E.g.f. A(x) satisfies: A(x) = x*exp(A(x)/(1 - A(x))). - _Ilya Gutkovskiy_, Apr 04 2019

%F a(n) = n * (n-2)! * LaguerreL(n-2, 1, -n) with a(0) = 0 and a(1) = 1. - _G. C. Greubel_, Feb 23 2021

%p spec := [S,{C=Set(B),S=Prod(Z,C),B=Sequence(S,1<= card)},labeled]:

%p seq(combstruct[count](spec,size=n), n=0..20);

%p # Alternatively:

%p a := n -> `if`(n<2,n, n!*hypergeom([-n+2],[2],-n));

%p seq(simplify(a(n)), n=0..17); # _Peter Luschny_, Apr 20 2016

%t Table[If[0<=n<=1, n, (n-1)! Sum[(n^k Binomial[n-2, k-1])/k!, {k,n-1}]], {n,0,20}] (* _Michael De Vlieger_, Apr 20 2016 *)

%t Table[If[n<2, n, n*(n-2)!*LaguerreL[n-2, 1, -n]], {n, 0, 20}] (* _G. C. Greubel_, Feb 23 2021 *)

%o (Maxima)

%o a(n):=if n=1 then 1 else ((n-1)!*sum((n^k*binomial(n-2,k-1))/k!,k,1,n-1)); /* _Vladimir Kruchinin_, May 10 2011 */

%o (SageMath) [n if n<2 else n*factorial(n-2)*gen_laguerre(n-2, 1, -n) for n in (0..20)] # _G. C. Greubel_, Feb 23 2021

%o (Magma) [n lt 2 select n else n*Factorial(n-2)*Evaluate(LaguerrePolynomial(n-2, 1), -n): n in [0..20]]; // _G. C. Greubel_, Feb 23 2021

%K easy,nonn

%O 0,3

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000