A212599 -id:A212599 - cp's OEIS frontend

A116956 Number of functions f:{1,2,...,n}->{1,2,...,n} with odd cycles only.

1, 1, 3, 18, 157, 1800, 25551, 432376, 8494809, 190029888, 4768313275, 132626098176, 4049755214517, 134677876657792, 4845193429684167, 187490897290080000, 7765153170076158001, 342721890859339812864, 16058392049508837366771, 796093438190851834236928

Offset: 0

Vladeta Jovovic, Mar 30 2006

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A070896, A060281, A060435, A070896.

Cf. A212599.

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)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 20 2016

t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Range[0, 20]! CoefficientList[
Series[((1 + t)/(1 - t))^(1/2), {x, 0, 20}], x]  (* Geoffrey Critzer, Dec 07 2011 *)

E.g.f.: sqrt((1-LambertW(-x))/(1+LambertW(-x))).
Sum_{k=0..n} binomial(n,k)*a(k)*a(n-k) = 2*n^n, n>0. - Vladeta Jovovic, Oct 11 2007
a(n) ~ n! * 2^(3/4)*Gamma(3/4)*exp(n)/(2*Pi*n^(3/4)). - Vaclav Kotesovec, Sep 24 2013

A246951 Number of endofunctions f on [n] such that the number of cycles of f is a multiple of 3 for each size that is a multiple of 3.

Original entry on oeis.org

1, 1, 4, 25, 224, 2625, 37936, 651553, 12963840, 293219361, 7429984000, 208486630121, 6417388432896, 214990745268065, 7787303908091904, 303250854519320625, 12633932936000045056, 560712315504995304897, 26410168773292930375680, 1315770870814835066545081

Offset: 0

Alois P. Heinz, Sep 08 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..380

Cf. A212599.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(j, igcd(i, 3))<>0, 0, b(n-i*j, i-1)*
      (i-1)!^j*multinomial(n, n-i*j, i$j)/j!), j=0..n/i)))
    end:
a:= n-> add(b(j$2)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

cp's OEIS Frontend

A116956 Number of functions f:{1,2,...,n}->{1,2,...,n} with odd cycles only.

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