%I #53 Dec 06 2024 08:45:41

%S 1,2,2,9,12,6,64,96,72,24,625,1000,900,480,120,7776,12960,12960,8640,

%T 3600,720,117649,201684,216090,164640,88200,30240,5040,2097152,

%U 3670016,4128768,3440640,2150400,967680,282240,40320,43046721

%N Number of endofunctions on n labeled points constructed from k rooted trees.

%C T(n,k) = number of endofunctions with k recurrent elements. - _Mitch Harris_, Jul 06 2006

%C The sum of row n is n^n, for any n. Basically the same sequence arises when studying random mappings (see A243203, A243202). - _Stanislav Sykora_, Jun 01 2014

%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 87, see (2.3.28).

%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 3.3.32.

%H Alois P. Heinz, <a href="/A066324/b066324.txt">Rows n = 1..141, flattened</a>

%F T(n,k) = k*n^(n-k)*(n-1)!/(n-k)!.

%F E.g.f. (relative to x): A(x, y)=1/(1-y*B(x)) - 1 = y*x +(2*y+2*y^2)*x^2/2! + (9*y+12*y^2+6*y^3)*x^3/3! + ..., where B(x) is e.g.f. A000169.

%F From _Peter Bala_, Sep 30 2011: (Start)

%F Let F(x,t) = x/(1+t*x)*exp(-x/(1+t*x)) = x*(1 - (1+t)*x + (1+4*t+2*t^2)*x^2/2! - ...). F is essentially the e.g.f. for A144084 (see also A021010). Then the e.g.f. for the present table is t*F(x,t)^(-1), where the compositional inverse is taken with respect to x.

%F Removing a factor of n from the n-th row entries results in A122525 in row reversed form.

%F (End)

%F Sum_{k=2..n} (k-1) * T(n,k) = A001864(n). - _Geoffrey Critzer_, Aug 19 2013

%F Sum_{k=1..n} k * T(n,k) = A063169(n). - _Alois P. Heinz_, Dec 15 2021

%e Triangle T(n,k) begins:

%e 1;

%e 2, 2;

%e 9, 12, 6;

%e 64, 96, 72, 24;

%e 625, 1000, 900, 480, 120;

%e 7776, 12960, 12960, 8640, 3600, 720;

%e 117649, 201684, 216090, 164640, 88200, 30240, 5040;

%e ...

%p T:= (n, k)-> k*n^(n-k)*(n-1)!/(n-k)!:

%p seq(seq(T(n, k), k=1..n), n=1..10); # _Alois P. Heinz_, Aug 22 2012

%t f[list_] := Select[list, # > 0 &]; t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Flatten[Map[f, Drop[Range[0, 10]! CoefficientList[Series[1/(1 - y*t), {x, 0, 10}], {x, y}], 1]]] (* _Geoffrey Critzer_, Dec 05 2011 *)

%o (PARI) T(n, k)=k*n^(n-k)*(n-1)!/(n-k)! \\ _Charles R Greathouse IV_, Dec 05 2011

%Y Column 1: A000169.

%Y Main diagonal: A000142.

%Y T(n, n-1): A062119.

%Y Row sums give A000312.

%Y Cf. A001864, A021010, A063169, A122525, A144084, A243203.

%K nonn,tabl,changed

%O 1,2

%A _Christian G. Bower_, Dec 14 2001