A052857 A simple grammar. a(n)=n*A052873(n-1).

0, 1, 2, 15, 184, 3145, 68976, 1846999, 58413440, 2130740721, 88061420800, 4066862460991, 207556068584448, 11600364266112505, 704664527894104064, 46226086991634882375, 3256882066245640093696, 245279323467051422886241

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 825
• Index entries for sequences related to Laguerre polynomials

Programs

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

Maple

spec := [S,{C=Set(B),S=Prod(Z,C),B=Sequence(S,1<= card)},labeled]:
seq(combstruct[count](spec,size=n), n=0..20);
# Alternatively:
a := n -> `if`(n<2,n, n!*hypergeom([-n+2],[2],-n));
seq(simplify(a(n)), n=0..17); # Peter Luschny, Apr 20 2016

Mathematica

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

Maxima

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 */

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

Formula

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

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

a(n) = n!*hypergeom([-n+2], [2], -n) for n>=2. - Peter Luschny, Apr 20 2016

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

E.g.f. A(x) satisfies: A(x) = x*exp(A(x)/(1 - A(x))). - Ilya Gutkovskiy, Apr 04 2019

a(n) = n * (n-2)! * LaguerreL(n-2, 1, -n) with a(0) = 0 and a(1) = 1. - G. C. Greubel, Feb 23 2021