A245049 Number A(n,k) of hybrid k-ary trees with n internal nodes; square array A(n,k), n>=0, k>=1, read by antidiagonals.

1, 1, 2, 1, 2, 3, 1, 2, 7, 5, 1, 2, 11, 31, 8, 1, 2, 15, 81, 154, 13, 1, 2, 19, 155, 684, 820, 21, 1, 2, 23, 253, 1854, 6257, 4575, 34, 1, 2, 27, 375, 3920, 24124, 60325, 26398, 55, 1, 2, 31, 521, 7138, 66221, 331575, 603641, 156233, 89, 1, 2, 35, 691, 11764, 148348, 1183077, 4736345, 6210059, 943174, 144

Offset: 0

Author

Alois P. Heinz, Jul 10 2014

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
   1,    1,     1,      1,       1,       1,       1, ...
   2,    2,     2,      2,       2,       2,       2, ...
   3,    7,    11,     15,      19,      23,      27, ...
   5,   31,    81,    155,     253,     375,     521, ...
   8,  154,   684,   1854,    3920,    7138,   11764, ...
  13,  820,  6257,  24124,   66221,  148348,  290305, ...
  21, 4575, 60325, 331575, 1183077, 3262975, 7585749, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• SeoungJi Hong and SeungKyung Park, Hybrid d-ary trees and their generalization, Bull. Korean Math. Soc. 51 (2014), No. 1, pp. 229-235

Crossrefs

Columns k=1-10 give: A000045(n+2), A007863, A215654, A239107, A239108, A239109, A245050, A245051, A245052, A245053.

Rows n=0-2 give: A000012, A007395, A004767(k-1).

Main diagonal gives A245054.

Programs

Maple

A:= (n, k)-> add(binomial((k-1)*n+i, i)*
    binomial((k-1)*n+i+1, n-i), i=0..n)/((k-1)*n+1):
seq(seq(A(n, 1+d-n), n=0..d), d=0..12);

Mathematica

A[n_, k_] := Sum[Binomial[(k-1)*n+i, i]*Binomial[(k-1)*n+i+1, n-i], {i, 0, n}]/((k-1)*n+1); Table[A[n, 1+d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 18 2017, translated from Maple *)

Formula

A(n,k) = 1/((k-1)*n+1) * Sum_{i=0..n} C((k-1)*n+i,i)*C((k-1)*n+i+1,n-i).

A(n,k) = [x^n] ((1+x)/(1-x-x^2))^((k-1)*n+1) / ((k-1)*n+1).

G.f. for column k satisfies: A_k(x) = (1+x*A_k(x)^(k-1)) * (1+x*A_k(x)^k).