A245406 Number of endofunctions on [n] such that no element has a preimage of cardinality two.

1, 1, 2, 9, 76, 825, 10206, 143521, 2313200, 42482313, 875799550, 19972186311, 498430219464, 13509979971241, 395352049852046, 12425644029361725, 417456939168255616, 14929305882415781265, 566234625018001351230, 22701936510037394021395, 959341639105178919209000

Offset

0,3

Links

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..396 (first 200 terms from Alois P. Heinz)

Formula

a(n) = n! * [x^n] (exp(x)-x^2/2!)^n.

a(n) ~ c * d^n * n^n / exp(n), where d = 2.166383277092734585444028653747119..., c = 0.8627963719760750933657356839596... . - Vaclav Kotesovec, Jul 24 2014

Maple

b:= proc(n, i) option remember; `if`(n=0 and i=0, 1, `if`(i<1, 0,
       add(`if`(j=2, 0, b(n-j, i-1) *binomial(n, j)), j=0..n)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);

Mathematica

Table[n!*SeriesCoefficient[(E^x - x^2/2)^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 23 2014 *)
With[{k=2}, Flatten[{1, Table[Sum[Binomial[n, j]*Binomial[n, k*j]*(-1)^j*(n-j)^(n-k*j)*(k*j)!/(k!)^j, {j, 0, n/k}], {n, 1, 20}]}]]  (* Vaclav Kotesovec, Jul 24 2014 *)

Crossrefs

Column k=2 of A245405.

Cf. A245493.

Sequence in context: A357539 A277181 A105785 * A337558 A276742 A123680

Adjacent sequences: A245403 A245404 A245405 * A245407 A245408 A245409

Keyword

nonn

Author

Alois P. Heinz, Jul 21 2014

Status

approved