A243202 -id:A243202 - cp's OEIS frontend

A066324 Number of endofunctions on n labeled points constructed from k rooted trees.

1, 2, 2, 9, 12, 6, 64, 96, 72, 24, 625, 1000, 900, 480, 120, 7776, 12960, 12960, 8640, 3600, 720, 117649, 201684, 216090, 164640, 88200, 30240, 5040, 2097152, 3670016, 4128768, 3440640, 2150400, 967680, 282240, 40320, 43046721

Offset: 1

Christian G. Bower, Dec 14 2001

T(n,k) = number of endofunctions with k recurrent elements. - Mitch Harris, Jul 06 2006

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

			Triangle T(n,k) begins:
       1;
       2,      2;
       9,     12,      6;
      64,     96,     72,     24;
     625,   1000,    900,    480,   120;
    7776,  12960,  12960,   8640,  3600,   720;
  117649, 201684, 216090, 164640, 88200, 30240, 5040;
  ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 87, see (2.3.28).
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 3.3.32.

Alois P. Heinz, Rows n = 1..141, flattened

Column 1: A000169.
Main diagonal: A000142.
T(n, n-1): A062119.
Row sums give A000312.
Cf. A001864, A021010, A063169, A122525, A144084, A243203.

T:= (n, k)-> k*n^(n-k)*(n-1)!/(n-k)!:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 22 2012

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 *)

T(n, k)=k*n^(n-k)*(n-1)!/(n-k)! \\ Charles R Greathouse IV, Dec 05 2011

T(n,k) = k*n^(n-k)*(n-1)!/(n-k)!.
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.
From Peter Bala, Sep 30 2011: (Start)
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.
Removing a factor of n from the n-th row entries results in A122525 in row reversed form.
(End)
Sum_{k=2..n} (k-1) * T(n,k) = A001864(n). - Geoffrey Critzer, Aug 19 2013
Sum_{k=1..n} k * T(n,k) = A063169(n). - Alois P. Heinz, Dec 15 2021

A243203 Terms of a particular integer decomposition of N^N.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 0, 9, 12, 6, 0, 64, 96, 72, 24, 0, 625, 1000, 900, 480, 120, 0, 7776, 12960, 12960, 8640, 3600, 720, 0, 117649, 201684, 216090, 164640, 88200, 30240, 5040, 0, 2097152, 3670016, 4128768, 3440640, 2150400, 967680, 282240, 40320, 0

Offset: 0

Stanislav Sykora, Jun 01 2014

a(n) is an element in the triangle of terms t(N,j) = c(N,j)*binomial(N,j), N = 0,1,2,3,... denoting a row, and j = 0,1,2,...r. The coefficients c(N,j) are specified numerically by the formula below. Note that all rows start with 0, which makes them easily recognizable.
The sum of every row is N^N.
Though the original contexts are different, this triangle matches that of A066324 except for row 0, and for the zero term of each row. On this point, see the comment in A243202.

			The first rows of the triangle are (first item is the row number N):
0 0
1 0, 1
2 0, 2, 2
3 0, 9, 12, 6
4 0, 64, 96, 72, 24
5 0, 625, 1000, 900, 480, 120
6 0, 7776, 12960, 12960, 8640, 3600, 720
7 0, 117649, 201684, 216090, 164640, 88200, 30240, 5040
8 0, 2097152, 3670016, 4128768, 3440640, 2150400, 967680, 282240, 40320

Stanislav Sykora, Table of n, a(n) for rows 0..100, flattened
S. Sykora, A Random Mapping Statistics and a Related Identity, Stan's Library, Volume V, June 2014.

Cf. A066324, A243202.

A243202(maxrow) = {
  my(v,n,j,irow,f);v = vector((maxrow+1)*(maxrow+2)/2);
  for(n=1,maxrow,irow=1+n*(n+1)/2;v[irow]=0;f=1;
  for(j=1,n,f *= j;v[irow+j] = j*f*n^(n-j-1)*binomial(n,j);););
  return(v);}

c(N,j)=N^(N-j)*(j/N)*j! for N>0 and 0<=j<=N, and c(N,j)=0 otherwise.

cp's OEIS Frontend

A066324 Number of endofunctions on n labeled points constructed from k rooted trees.

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