A231602 Triangular array read by rows: T(n,k) is the number of rooted labeled trees on n nodes that have exactly k nodes with outdegree = 1, n>=1, 0<=k<=n-1.

1, 0, 2, 3, 0, 6, 4, 36, 0, 24, 65, 80, 360, 0, 120, 306, 1950, 1200, 3600, 0, 720, 4207, 12852, 40950, 16800, 37800, 0, 5040, 38424, 235592, 359856, 764400, 235200, 423360, 0, 40320, 573057, 2766528, 8481312, 8636544, 13759200, 3386880, 5080320, 0, 362880

Offset: 1

Author

Geoffrey Critzer, Nov 11 2013

Keywords

• nonn
• tabl

Comments

T(n,k) is also the number of functions f:{1,2,...,n-1}->{1,2,...,n} that have exactly k elements whose preimage has cardinality = 1.

T(n,n-1) = n! = A000142(n).

Column k = 0 = A060356(n).

Row sums = n^(n-1) = A000169(n).

Refinement given by A248120. Sum coefficients of the partition polynomials with h_1 = (1') = t and all other h_n = (n') = 1 to obtain this entry. - Tom Copeland, Feb 01 2016

Examples

			1;
0, 2;
3, 0, 6;
4, 36, 0, 24;
65, 80, 360, 0, 120;
306, 1950, 1200, 3600, 0, 720;
4207, 12852, 40950, 16800, 37800, 0, 5040;
38424, 235592, 359856, 764400, 235200, 423360, 0, 40320;
....0..........0........
....|........./ \.......
....0........0...0......
.../ \.......|..........
..0   0......0..........
T(4,1) = 36.  Both of these graphs on 4 nodes have exactly 1 node that has outdegree = 1.  There are 12 + 24 = 36 labelings.
		

Links

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

Crossrefs

Cf. A055302, A206823.

Cf. A248120.

Programs

Maple

with(combinat): C:= binomial:
b:= proc(t, i, u) option remember; `if`(t=0, 1,
      `if`(i<2, 0, b(t, i-1, u) +add(multinomial(t, t-i*j, i$j)
      *b(t-i*j, i-1, u-j)*u!/(u-j)!/j!, j=1..t/i)))
    end:
T:= (n, k)-> C(n, k)*C(n-1, k)*k! *b(n-1-k$2, n-k):
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Nov 12 2013

Mathematica

nn=8;Table[Table[Drop[Range[0,nn]!CoefficientList[Series[-ProductLog[x/(-1-x+x y)],{x,0,nn}],{x,y}],1][[r,c]],{c,1,r}],{r,1,nn}]//Grid

Formula

E.g.f. satisfies A(x,y) = y*x*A(x,y) + x*( exp(A(x,y)) - A(x,y) ).