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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. 11
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

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).

EXAMPLE

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, ...

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-Fran├žois Alcover, Feb 18 2017, translated from Maple *)

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.

Sequence in context: A263703 A263752 A101161 * A214261 A097825 A002343

Adjacent sequences:  A245046 A245047 A245048 * A245050 A245051 A245052

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 10 2014

STATUS

approved

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Last modified October 20 11:18 EDT 2019. Contains 328257 sequences. (Running on oeis4.)