A245407 Number of endofunctions on [n] such that no element has a preimage of cardinality three.

1, 1, 4, 24, 208, 2325, 31956, 520723, 9812160, 209843145, 5020469200, 132844628411, 3851705048016, 121428210575581, 4135403154270584, 151297710936948675, 5917989635505922816, 246444213949305536017, 10885732208011517726880, 508350675616737391265563

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

a(n) = n! * [x^n] (exp(x)-x^3/3!)^n.

a(n) ~ c * d^n * n^n / exp(n), where d = 2.52566039645910026750819504865..., c = 1.031458655073968039932844239... . - 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=3, 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^3/6)^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 23 2014 *)
With[{k=3}, 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=3 of A245405.

Sequence in context: A368267 A297218 A010039 * A162314 A369723 A323869

Adjacent sequences: A245404 A245405 A245406 * A245408 A245409 A245410

Keyword

nonn

Author

Alois P. Heinz, Jul 21 2014

Status

approved