OFFSET
1,2
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. An element that is not recurrent is a nonrecurrent element.
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
FORMULA
E.g.f.: 1/(1-x*exp(A(x,y))), where A(x,y) = Sum_{n>=1} n^(n-1)*(y*x)^n/n!.
EXAMPLE
T(2,1) = 2 because we have 1->1 2->1; and 1->2 2->2.
: 1;
: 2, 2;
: 6, 12, 9;
: 24, 72, 96, 64;
: 120, 480, 900, 1000, 625;
: 720, 3600, 8640, 12960, 12960, 7776;
: 5040, 30240, 88200, 164640, 216090, 201684, 117649;
MAPLE
b:= proc(n) option remember; `if`(n=0, 1, add(
(j-1)!*b(n-j)*binomial(n-1, j-1), j=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(add(
b(j)*(x*n)^(n-j)*binomial(n-1, j-1), j=0..n)):
seq(T(n), n=1..10); # Alois P. Heinz, May 22 2016
MATHEMATICA
nn=8; f[list_]:=Select[list, #>0&]; t=Sum[n^(n-1)x^n y^n/n!, {n, 1, nn}]; Drop[Map[f, Range[0, nn]!CoefficientList[Series[1/(1-x Exp[t]), {x, 0, nn}], {x, y}]], 1]//Grid
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Nov 25 2012
STATUS
approved