A052804 - OEIS

%I #23 Dec 13 2023 08:17:30

%S 0,0,2,3,20,90,714,5460,54704,580608,7214040,96932880,1452396912,

%T 23507621280,414102201408,7827185489760,158757800613120,

%U 3429996441661440,78775916315263488,1914627403408320000,49126748261368331520,1326584986873331189760

%N A simple grammar: cycles of rooted cycles.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=765">Encyclopedia of Combinatorial Structures 765</a>

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

%F E.g.f.: -log(1 + x*log(1-x)). - _Arkadiusz Wesolowski_, Feb 21 2013

%F a(n) ~ (n-1)! * r^n, where r = 1.349976485401125... is the root of the equation (r-1)*exp(r) = r. - _Vaclav Kotesovec_, Oct 01 2013

%F a(n) =  n! * Sum_{k=1..floor(n/2)}(k-1)! * |Stirling1(n-k,k)|/(n-k)!. - _Seiichi Manyama_, Dec 13 2023

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

%t nn = 25; Range[0, nn]! CoefficientList[Series[Log[-1/(-1 + Log[-1/(-1 + x)]*x)], {x, 0, nn}], x] (* _T. D. Noe_, Feb 21 2013 *)

%o (PARI)

%o N = 66;  x = 'x + O('x^N);

%o egf = -log(1 + x*log(1-x)) + 'c0;

%o gf = serlaplace(egf);

%o v = Vec(gf);  v[1]-='c0;  v

%o /* _Joerg Arndt_, Feb 21 2013 */

%Y Cf. A052830, A052858.

%K easy,nonn

%O 0,3

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