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.

1, 1, 3, 12, 73, 580, 5601, 63994, 844929, 12647016, 211616065, 3914510446, 79320037281, 1747219469164, 41569414869633, 1062343684252530, 29023112392093441, 844101839207139280, 26038508978625589377, 849150487829425227094, 29189561873274715264545

Offset: 0

Author

Geoffrey Critzer, Jul 25 2016

Keywords

• nonn

Comments

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.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..411

Crossrefs

Cf. A000246, A006153, A009444, A116956, A216401.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
       (j-1)!*b(n-j)*binomial(n-1, j-1), 0), j=1..n))
    end:
a:= n-> add(b(j)*j^(n-j)*binomial(n, j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 25 2016

Mathematica

nn = 20; Range[0, nn]! CoefficientList[Series[Sqrt[(1 + z*Exp[z])/(1 - z*Exp[z])], {z, 0, nn}], z]

PARI

default(seriesprecision, 30);
S=sqrt((1 + x*exp(x))/(1 - x*exp(x)));
v=Vec(S); for(n=2,#v-1,v[n+1]*=n!); v \\ Charles R Greathouse IV, Jul 29 2016

Formula

E.g.f.: sqrt((1 + z*exp(z))/(1 - z*exp(z))).

Exponential transform of A216401.

a(n) ~ 2 * n^n / (sqrt(1+LambertW(1)) * LambertW(1)^n * exp(n)). - Vaclav Kotesovec, Jun 26 2022