A034781 Triangle of number of rooted trees with n >= 2 nodes and height h >= 1.
1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 10, 18, 13, 5, 1, 1, 14, 38, 36, 19, 6, 1, 1, 21, 76, 93, 61, 26, 7, 1, 1, 29, 147, 225, 180, 94, 34, 8, 1, 1, 41, 277, 528, 498, 308, 136, 43, 9, 1, 1, 55, 509, 1198, 1323, 941, 487, 188, 53, 10, 1
Offset: 2
Examples
Triangle begins: 1; 1 1; 1 2 1; 1 4 3 1; 1 6 8 4 1; 1 10 18 13 5 1; 1 14 38 36 19 6 1; thus there are 10 trees with 7 nodes and height 2.
Links
- Alois P. Heinz, Rows n = 2..142, flattened
- Marko Riedel, Counting the number of rooted trees of a certain height
- Marko Riedel, Maple code for sequence (OGF)
- 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)
- Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 10 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
- Index entries for sequences related to rooted trees
- Index entries for sequences related to trees
Crossrefs
Programs
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Maple
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0, add(binomial(b((i-1)$2, k-1)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i))) end: T:= (n, k)-> b((n-1)$2, k) -b((n-1)$2, k-1): seq(seq(T(n, k), k=1..n-1), n=2..16); # Alois P. Heinz, Jul 31 2013 -
Mathematica
Drop[Map[Select[#, # > 0 &] &, Transpose[ Prepend[Table[ f[n_] := Nest[CoefficientList[ Series[Product[1/(1 - x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 10}], x] &, {1}, n]; f[m] - f[m - 1], {m, 2, 10}], Prepend[Table[1, {10}], 0]]]], 1] // Grid (* Geoffrey Critzer, Aug 01 2013 *) b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1 || k<1, 0, Sum[Binomial[b[i-1, i-1, k-1]+j-1, j]*b[n-i*j, i-1, k], {j, 0, n/i}]]]; T[n_, k_] := b[n-1, n-1, k]-b[n-1, n-1, k-1]; Table[T[n, k], {n, 2, 16}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 11 2014, after Alois P. Heinz *) -
Python
def A034781(n, k): return A375467(n, k) - A375467(n, k - 1) for n in range(2, 10): print([A034781(n, k) for k in range(2, n + 1)]) # Peter Luschny, Aug 30 2024
Formula
Reference gives recurrence.
Extensions
More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Sep 19 2003