A299038 Number A(n,k) of rooted trees with n nodes where each node has at most k children; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 3, 1, 0, 1, 1, 1, 2, 4, 6, 1, 0, 1, 1, 1, 2, 4, 8, 11, 1, 0, 1, 1, 1, 2, 4, 9, 17, 23, 1, 0, 1, 1, 1, 2, 4, 9, 19, 39, 46, 1, 0, 1, 1, 1, 2, 4, 9, 20, 45, 89, 98, 1, 0, 1, 1, 1, 2, 4, 9, 20, 47, 106, 211, 207, 1, 0

Offset

0,19

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Formula

A(n,k) = Sum_{i=0..k} A244372(n,i) for n>0, A(0,k) = 1.

Example

Square array A(n,k) begins:
  1, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  1, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0, 1,   2,   2,   2,   2,   2,   2,   2,   2,   2, ...
  0, 1,   3,   4,   4,   4,   4,   4,   4,   4,   4, ...
  0, 1,   6,   8,   9,   9,   9,   9,   9,   9,   9, ...
  0, 1,  11,  17,  19,  20,  20,  20,  20,  20,  20, ...
  0, 1,  23,  39,  45,  47,  48,  48,  48,  48,  48, ...
  0, 1,  46,  89, 106, 112, 114, 115, 115, 115, 115, ...
  0, 1,  98, 211, 260, 277, 283, 285, 286, 286, 286, ...
  0, 1, 207, 507, 643, 693, 710, 716, 718, 719, 719, ...

Maple

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
A:= (n, k)-> `if`(n=0, 1, b(n-1$2, k$2)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[ b[i-1, i-1, k, k]+j-1, j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]];
A[n_, k_] := If[n == 0, 1, b[n - 1, n - 1, k, k]];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 04 2018, from Maple *)

Prog

(Python)
from sympy import binomial
from sympy.core.cache import cacheit
@cacheit
def b(n, i, t, k): return 1 if n==0 else 0 if i<1 else sum([binomial(b(i-1, i-1, k, k)+j-1, j)*b(n-i*j, i-1, t-j, k) for j in range(min(t, n//i)+1)])
def A(n, k): return 1 if n==0 else b(n-1, n-1, k, k)
for d in range(15): print([A(n, d-n) for n in range(d+1)]) # Indranil Ghosh, Mar 02 2018, after Maple code

Crossrefs

Columns k=1-11 give: A000012, A001190(n+1), A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556.

Main diagonal gives A000081 for n>0.

A(2n,n) gives A299039.

Cf. A244372.

Sequence in context: A003137 A353048 A006842 * A273693 A219967 A060505

Adjacent sequences: A299035 A299036 A299037 * A299039 A299040 A299041

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 01 2018

Status

approved