cp's OEIS Frontend

A275385 Number of labeled functional digraphs on n nodes with only odd sized cycles and such that every vertex is at a distance of at most 1 from a cycle.

%I A275385 #26 Jun 26 2022 05:48:00
%S A275385 1,1,3,12,73,580,5601,63994,844929,12647016,211616065,3914510446,
%T A275385 79320037281,1747219469164,41569414869633,1062343684252530,
%U A275385 29023112392093441,844101839207139280,26038508978625589377,849150487829425227094,29189561873274715264545
%N A275385 Number of labeled functional digraphs on n nodes with only odd sized cycles and such that every vertex is at a distance of at most 1 from a cycle.
%C A275385 Equivalently, these are the functions counted by A116956 with the additional constraint that every element is mapped to a recurrent element.  A recurrent element is an element on a cycle in the functional digraph.
%H A275385 Alois P. Heinz, <a href="/A275385/b275385.txt">Table of n, a(n) for n = 0..411</a>
%F A275385 E.g.f.: sqrt((1 + z*exp(z))/(1 - z*exp(z))).
%F A275385 Exponential transform of A216401.
%F A275385 a(n) ~ 2 * n^n / (sqrt(1+LambertW(1)) * LambertW(1)^n * exp(n)). - _Vaclav Kotesovec_, Jun 26 2022
%p A275385 b:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
%p A275385        (j-1)!*b(n-j)*binomial(n-1, j-1), 0), j=1..n))
%p A275385     end:
%p A275385 a:= n-> add(b(j)*j^(n-j)*binomial(n, j), j=0..n):
%p A275385 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jul 25 2016
%t A275385 nn = 20; Range[0, nn]! CoefficientList[Series[Sqrt[(1 + z*Exp[z])/(1 - z*Exp[z])], {z, 0, nn}], z]
%o A275385 (PARI) default(seriesprecision, 30);
%o A275385 S=sqrt((1 + x*exp(x))/(1 - x*exp(x)));
%o A275385 v=Vec(S); for(n=2,#v-1,v[n+1]*=n!); v \\ _Charles R Greathouse IV_, Jul 29 2016
%Y A275385 Cf. A000246, A006153, A009444, A116956, A216401.
%K A275385 nonn
%O A275385 0,3
%A A275385 _Geoffrey Critzer_, Jul 25 2016