A034855 Triangle read by rows giving number of rooted labeled trees with n >= 2 nodes and height d >= 1.

2, 3, 6, 4, 36, 24, 5, 200, 300, 120, 6, 1170, 3360, 2520, 720, 7, 7392, 38850, 43680, 22680, 5040, 8, 50568, 475776, 757680, 551040, 221760, 40320, 9, 372528, 6231960, 13747104, 12836880, 7136640, 2358720, 362880, 10, 2936070, 87530400, 264181680

Offset

2,1

Links

Alois P. Heinz, Rows n = 2..101, flattened

Marko Riedel, Counting the number of rooted trees of a certain height

Marko Riedel, Maple code for sequence (EGF)

J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478. [broken link]

J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.

J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)

Index entries for sequences related to trees

Formula

Riordan reference gives recurrence.

Example

2;
3,    6;
4,   36,    24;
5,  200,   300,   120;
6, 1170,  3360,  2520,   720;
7, 7392, 38850, 43680, 22680, 5040;

Maple

gf:= proc(k) gf(k):= `if`(k=0, x, x*exp(gf(k-1))) end:
A:= proc(n, k) A(n, k):= n!*coeff(series(gf(k), x, n+1), x, n) end:
T:= (n, d)-> A(n, d) -A(n, d-1):
seq(seq(T(n, d), d=1..n-1), n=2..12);  # Alois P. Heinz, Sep 21 2012

Mathematica

gf[k_] := gf[k] = If[k == 0, x, x*E^gf[k - 1]]; a[n_, k_] := n!*Coefficient[ Series[gf[k], {x, 0, n + 1}], x, n]; t[n_, d_] := a[n, d] - a[n, d - 1]; Table[t[n, d], {n, 2, 12}, {d, 1, n - 1}] // Flatten (* Jean-François Alcover, Jan 15 2013, translated from Alois P. Heinz's Maple program *)

Crossrefs

Cf. A001854, A234953, A000435, A236396.

Sequence in context: A096357 A091507 A098282 * A105214 A358940 A136315

Adjacent sequences: A034852 A034853 A034854 * A034856 A034857 A034858

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 27 2004

Status

approved