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

Author

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

Keywords

• nonn
• easy

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

Crossrefs

Cf. A353880, A353881, A353882.

Cf. A052848, A066166.

Programs

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 *)

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

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

Extensions

More terms from Alois P. Heinz, Mar 16 2016