A236396 Triangle read by rows: T(n,k) = number of rooted labeled trees with n nodes and height <= k, for n >= 1, 0 <= k <= n-1.

1, 0, 2, 0, 3, 9, 0, 4, 40, 64, 0, 5, 205, 505, 625, 0, 6, 1176, 4536, 7056, 7776, 0, 7, 7399, 46249, 89929, 112609, 117649, 0, 8, 50576, 526352, 1284032, 1835072, 2056832, 2097152, 0, 9, 372537, 6604497, 20351601, 33188481, 40325121, 42683841, 43046721

Offset

1,3

Comments

If we replace each row by its differences we get A034855.

Links

Alois P. Heinz, Rows n = 1..100, flattened

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.

Example

Triangle begins:
[1],
[0, 2],
[0, 3, 9],
[0, 4, 40, 64],
[0, 5, 205, 505, 625],
[0, 6, 1176, 4536, 7056, 7776],
[0, 7, 7399, 46249, 89929, 112609, 117649],
[0, 8, 50576, 526352, 1284032, 1835072, 2056832, 2097152],
...

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:
[seq([seq(A(n, d), d=0..n-1)], n=1..12)];

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]; Table[Table[a[n, d], {d, 0, n-1}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Mar 07 2014, after Maple *)

Crossrefs

Cf. A034855, A001854, A235595, A234953.

Sequence in context: A071411 A336326 A255384 * A251592 A365985 A327244

Adjacent sequences: A236393 A236394 A236395 * A236397 A236398 A236399

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 28 2014

Status

approved