|
| |
|
|
A215720
|
|
The number of functions f:{1,2,...,n}->{1,2,...,n}, endofunctions, such that exactly one nonrecurrent element is mapped into each recurrent element.
|
|
1
|
|
|
|
1, 0, 2, 6, 60, 560, 7350, 111552, 2009672, 41378976, 963527850, 25009038560, 716437784172, 22453784964624, 764345507271710, 28085186967504240, 1107971902218683280, 46710909213378892352, 2095883952368863510098, 99724281567446320231104, 5015524096516005263567540
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
x in {1,2,...,n} is a recurrent element if there is some k such that f^k(x) = x where f^k(x) denotes iterated functional composition. In other words, a recurrent element is in a cycle of the functional digraph.
|
|
|
LINKS
|
|
|
|
FORMULA
|
E.g.f.: 1/(1 - x*T(x)) where T(x) is the e.g.f. for A000169.
a(n) = n! * Sum_{i=0..floor(n/2)} i*(n-i)^(n-2*i-1)/(n-2*i)! for n>0, a(0) = 1. - Alois P. Heinz, Aug 22 2012
|
|
|
EXAMPLE
|
a(2) = 2 because we have: (1->1,2->1), (1->2,2->2).
|
|
|
MAPLE
|
a:= n-> `if`(n=0, 1, n! *add(i*(n-i)^(n-2*i-1)/(n-2*i)!, i=0..n/2)):
|
|
|
MATHEMATICA
|
nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
Range[0, nn]! CoefficientList[Series[1/(1 - x t) , {x, 0, nn}], x]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
