%I #48 Dec 04 2023 06:36:46

%S 1,0,2,3,32,150,1524,12600,147328,1705536,23681520,345605040,

%T 5654922624,98624766240,1870594556544,37794037488480,817362198512640,

%U 18742996919324160,455648694329309184,11683777530785978880,315505598702787118080,8943481464393674096640

%N A simple grammar: sequences of rooted cycles.

%C Asymptotic behavior (formula 3.2.) in the INRIA reference is wrong! - _Vaclav Kotesovec_, Jun 03 2019

%H Seiichi Manyama, <a href="/A052830/b052830.txt">Table of n, a(n) for n = 0..428</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=795">Encyclopedia of Combinatorial Structures 795</a>

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

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

%F 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

%F a(0) = 1; a(n) = n! * Sum_{k=2..n} 1/(k-1) * a(n-k)/(n-k)!. - _Seiichi Manyama_, May 04 2022

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

%t CoefficientList[Series[1/(1+x*Log[1-x]), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)

%o (Maxima) a(n):=(-1)^(n)*n!*sum((k!*stirling1(n-k,k))/(n-k)!,k,0,n/2); /* _Vladimir Kruchinin_, Nov 16 2011 */

%o (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

%o (PARI) a(n) = n!*sum(k=0, n\2, k!*abs(stirling(n-k, k, 1))/(n-k)!); \\ _Seiichi Manyama_, May 04 2022

%Y Cf. A353880, A353881, A353882.

%Y Cf. A052848, A066166.

%K nonn,easy

%O 0,3

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

%E More terms from _Alois P. Heinz_, Mar 16 2016