A052830 A simple grammar: sequences of rooted cycles.

1, 0, 2, 3, 32, 150, 1524, 12600, 147328, 1705536, 23681520, 345605040, 5654922624, 98624766240, 1870594556544, 37794037488480, 817362198512640, 18742996919324160, 455648694329309184, 11683777530785978880, 315505598702787118080, 8943481464393674096640

Offset

0,3

Comments

Asymptotic behavior (formula 3.2.) in the INRIA reference is wrong! - Vaclav Kotesovec, Jun 03 2019

Links

Seiichi Manyama, Table of n, a(n) for n = 0..428

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 795

Formula

E.g.f.: 1/(1-x*log(1/(1-x))).

a(n) = (-1)^n*n!*Sum_{k=0..floor(n/2)} k!*Stirling1(n-k,k)/(n-k)!. - Vladimir Kruchinin, Nov 16 2011

a(n) ~ n! * r^(n+1)/(r+1/(r-1)), where r = 1.349976485401125... is the root of the equation (r-1)*exp(r) = r. - Vaclav Kotesovec, Sep 30 2013

a(0) = 1; a(n) = n! * Sum_{k=2..n} 1/(k-1) * a(n-k)/(n-k)!. - Seiichi Manyama, May 04 2022

Maple

spec := [S, {B=Prod(C, Z), C=Cycle(Z), S=Sequence(B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);

Mathematica

CoefficientList[Series[1/(1+x*Log[1-x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

Prog

(Maxima) a(n):=(-1)^(n)*n!*sum((k!*stirling1(n-k, k))/(n-k)!, k, 0, n/2); /* Vladimir Kruchinin, Nov 16 2011 */
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=2, i, 1/(j-1)*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, May 04 2022
(PARI) a(n) = n!*sum(k=0, n\2, k!*abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, May 04 2022

Crossrefs

Cf. A353880, A353881, A353882.

Cf. A052848, A066166.

Sequence in context: A107124 A141049 A354612 * A357265 A356904 A041895

Adjacent sequences: A052827 A052828 A052829 * A052831 A052832 A052833

Keyword

nonn,easy

Author

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

Extensions

More terms from Alois P. Heinz, Mar 16 2016

Status

approved