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A052857
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A simple grammar.
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2
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0, 1, 2, 15, 184, 3145, 68976, 1846999, 58413440, 2130740721, 88061420800, 4066862460991, 207556068584448, 11600364266112505, 704664527894104064, 46226086991634882375, 3256882066245640093696, 245279323467051422886241
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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MAPLE
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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));
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MATHEMATICA
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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 *)
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PROG
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(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
(Sage) [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
(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
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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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STATUS
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approved
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