A055302 Triangle of number of labeled rooted trees with n nodes and k leaves, n >= 1, 1 <= k <= n.

1, 2, 0, 6, 3, 0, 24, 36, 4, 0, 120, 360, 140, 5, 0, 720, 3600, 3000, 450, 6, 0, 5040, 37800, 54600, 18900, 1302, 7, 0, 40320, 423360, 940800, 588000, 101136, 3528, 8, 0, 362880, 5080320, 16087680, 15876000, 5143824, 486864, 9144, 9, 0, 3628800

Offset: 1

Author

Christian G. Bower, May 11 2000

Keywords

• nonn
• tabl
• eigen

Comments

Beginning with the second row, dividing each row by n gives the mirror of row n-1 of A141618. Under the exponential transform, the mirror of A141618 is generated, relating the number of connected graphs here to the number of disconnected graphs associated with A141618 (cf. A127671 and A036040). - Tom Copeland, Oct 25 2014

Examples

			Triangle begins
     1,
     2,     0;
     6,     3,     0;
    24,    36,     4,     0;
   120,   360,   140,     5,    0;
   720,  3600,  3000,   450,    6, 0;
  5040, 37800, 54600, 18900, 1302, 7, 0;
		

References

• Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 313.

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• N. J. A. Sloane, Transforms
• Index entries for sequences related to rooted trees

Crossrefs

Row sums give A000169. Columns 1 through 12: A000142, A055303-A055313. Cf. A055314.

Cf. A248120 for a natural refinement.

Programs

Maple

T:= (n, k)-> (n!/k!)*Stirling2(n-1, n-k):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Nov 13 2013

Mathematica

Table[Table[n!/k! StirlingS2[n-1,n-k], {k,1,n}], {n,0,10}]//Grid  (* Geoffrey Critzer, Dec 01 2012 *)

PARI

A055302(n,k)=n!/k!*stirling(n-1, n-k,2);
for(n=1,10,for(k=1,n,print1(A055302(n,k),", "));print());
\\ Joerg Arndt, Oct 27 2014

Formula

E.g.f. (relative to x) satisfies: A(x,y) = xy + x*exp(A(x,y)) - x. Divides by n and shifts up under exponential transform.

T(n,k) = (n!/k!)*Stirling2(n-1, n-k). - Vladeta Jovovic, Jan 28 2004

T(n,k) = A055314(n,k)*(n-k) + A055314(n,k+1)*(k+1). The first term is the number of such trees with root degree > 1 while the second term is the number of such trees with root degree = 1. This simplifies to the above formula by Vladeta Jovovic. - Geoffrey Critzer, Dec 01 2012

E.g.f.: G(x,t) = log[1 + t * N(x*t,1/t)], where N(x,t) is the e.g.f. of A141618. Also, G(x*t,1/t)= log[1 + N(x,t)/t] is the comp. inverse in x of x / [1 + t * (e^x - 1)]. - Tom Copeland, Oct 26 2014