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%I #20 Sep 08 2018 17:19:14
%S 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,
%T 21,1,2,23,253,1854,6257,4575,34,1,2,27,375,3920,24124,60325,26398,55,
%U 1,2,31,521,7138,66221,331575,603641,156233,89,1,2,35,691,11764,148348,1183077,4736345,6210059,943174,144
%N 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.
%H Alois P. Heinz, <a href="/A245049/b245049.txt">Antidiagonals n = 0..140, flattened</a>
%H SeoungJi Hong and SeungKyung Park, <a href="http://dx.doi.org/10.4134/BKMS.2014.51.1.229">Hybrid d-ary trees and their generalization</a>, Bull. Korean Math. Soc. 51 (2014), No. 1, pp. 229-235
%F 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).
%F A(n,k) = [x^n] ((1+x)/(1-x-x^2))^((k-1)*n+1) / ((k-1)*n+1).
%F G.f. for column k satisfies: A_k(x) = (1+x*A_k(x)^(k-1)) * (1+x*A_k(x)^k).
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 2, 2, 2, 2, 2, 2, 2, ...
%e 3, 7, 11, 15, 19, 23, 27, ...
%e 5, 31, 81, 155, 253, 375, 521, ...
%e 8, 154, 684, 1854, 3920, 7138, 11764, ...
%e 13, 820, 6257, 24124, 66221, 148348, 290305, ...
%e 21, 4575, 60325, 331575, 1183077, 3262975, 7585749, ...
%p A:= (n, k)-> add(binomial((k-1)*n+i, i)*
%p binomial((k-1)*n+i+1, n-i), i=0..n)/((k-1)*n+1):
%p seq(seq(A(n, 1+d-n), n=0..d), d=0..12);
%t 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 *)
%Y Columns k=1-10 give: A000045(n+2), A007863, A215654, A239107, A239108, A239109, A245050, A245051, A245052, A245053.
%Y Rows n=0-2 give: A000012, A007395, A004767(k-1).
%Y Main diagonal gives A245054.
%K nonn,tabl
%O 0,3
%A _Alois P. Heinz_, Jul 10 2014
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