A052804 A simple grammar: cycles of rooted cycles.

0, 0, 2, 3, 20, 90, 714, 5460, 54704, 580608, 7214040, 96932880, 1452396912, 23507621280, 414102201408, 7827185489760, 158757800613120, 3429996441661440, 78775916315263488, 1914627403408320000, 49126748261368331520, 1326584986873331189760

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 765

Crossrefs

Cf. A052830, A052858.

Programs

Maple

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

Mathematica

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 *)

PARI

N = 66;  x = 'x + O('x^N);
egf = -log(1 + x*log(1-x)) + 'c0;
gf = serlaplace(egf);
v = Vec(gf);  v[1]-='c0;  v
/* Joerg Arndt, Feb 21 2013 */

Formula

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

E.g.f.: -log(1 + x*log(1-x)). - Arkadiusz Wesolowski, Feb 21 2013

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

a(n) = n! * Sum_{k=1..floor(n/2)} (k-1)! * |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, Dec 13 2023

a(0) = a(1) = 0; a(n) = n * (n-2)! + Sum_{k=2..n-1} k * (k-2)! * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Jan 21 2025