cp's OEIS Frontend

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

%I A245406 #17 Jul 24 2014 08:50:50
%S A245406 1,1,2,9,76,825,10206,143521,2313200,42482313,875799550,19972186311,
%T A245406 498430219464,13509979971241,395352049852046,12425644029361725,
%U A245406 417456939168255616,14929305882415781265,566234625018001351230,22701936510037394021395,959341639105178919209000
%N A245406 Number of endofunctions on [n] such that no element has a preimage of cardinality two.
%H A245406 Alois P. Heinz and Vaclav Kotesovec, <a href="/A245406/b245406.txt">Table of n, a(n) for n = 0..396</a> (first 200 terms from Alois P. Heinz)
%F A245406 a(n) = n! * [x^n] (exp(x)-x^2/2!)^n.
%F A245406 a(n) ~ c * d^n * n^n / exp(n), where d = 2.166383277092734585444028653747119..., c = 0.8627963719760750933657356839596... . - _Vaclav Kotesovec_, Jul 24 2014
%p A245406 b:= proc(n, i) option remember; `if`(n=0 and i=0, 1, `if`(i<1, 0,
%p A245406        add(`if`(j=2, 0, b(n-j, i-1) *binomial(n, j)), j=0..n)))
%p A245406     end:
%p A245406 a:= n-> b(n$2):
%p A245406 seq(a(n), n=0..25);
%t A245406 Table[n!*SeriesCoefficient[(E^x - x^2/2)^n, {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 23 2014 *)
%t A245406 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 *)
%Y A245406 Column k=2 of A245405.
%Y A245406 Cf. A245493.
%K A245406 nonn
%O A245406 0,3
%A A245406 _Alois P. Heinz_, Jul 21 2014