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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.

%I A245049 #20 Sep 08 2018 17:19:14
%S A245049 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 A245049 21,1,2,23,253,1854,6257,4575,34,1,2,27,375,3920,24124,60325,26398,55,
%U A245049 1,2,31,521,7138,66221,331575,603641,156233,89,1,2,35,691,11764,148348,1183077,4736345,6210059,943174,144
%N 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.
%H A245049 Alois P. Heinz, <a href="/A245049/b245049.txt">Antidiagonals n = 0..140, flattened</a>
%H A245049 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 A245049 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 A245049 A(n,k) = [x^n] ((1+x)/(1-x-x^2))^((k-1)*n+1) / ((k-1)*n+1).
%F A245049 G.f. for column k satisfies: A_k(x) = (1+x*A_k(x)^(k-1)) * (1+x*A_k(x)^k).
%e A245049 Square array A(n,k) begins:
%e A245049    1,    1,     1,      1,       1,       1,       1, ...
%e A245049    2,    2,     2,      2,       2,       2,       2, ...
%e A245049    3,    7,    11,     15,      19,      23,      27, ...
%e A245049    5,   31,    81,    155,     253,     375,     521, ...
%e A245049    8,  154,   684,   1854,    3920,    7138,   11764, ...
%e A245049   13,  820,  6257,  24124,   66221,  148348,  290305, ...
%e A245049   21, 4575, 60325, 331575, 1183077, 3262975, 7585749, ...
%p A245049 A:= (n, k)-> add(binomial((k-1)*n+i, i)*
%p A245049     binomial((k-1)*n+i+1, n-i), i=0..n)/((k-1)*n+1):
%p A245049 seq(seq(A(n, 1+d-n), n=0..d), d=0..12);
%t A245049 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 A245049 Columns k=1-10 give: A000045(n+2), A007863, A215654, A239107, A239108, A239109, A245050, A245051, A245052, A245053.
%Y A245049 Rows n=0-2 give: A000012, A007395, A004767(k-1).
%Y A245049 Main diagonal gives A245054.
%K A245049 nonn,tabl
%O A245049 0,3
%A A245049 _Alois P. Heinz_, Jul 10 2014