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A245054 Number of hybrid (n+1)-ary trees with n internal nodes.

%I A245054 #11 Sep 08 2018 17:18:41
%S A245054 1,2,11,155,3920,148348,7585749,492007235,38798085127,3609589528430,
%T A245054 387451906370509,47166300422957938,6423902286587614629,
%U A245054 968148639856266236900,159999832729471473179245,28775750341340155354161817,5595702924360902427922341048
%N A245054 Number of hybrid (n+1)-ary trees with n internal nodes.
%H A245054 Alois P. Heinz, <a href="/A245054/b245054.txt">Table of n, a(n) for n = 0..200</a>
%H A245054 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 A245054 a(n) = 1/(n^2+1) * Sum_{i=0..n} C(n^2+i,i) * C(n^2+i+1,n-i).
%F A245054 a(n) = [x^n] ((1+x)/(1-x-x^2))^(n^2+1) / (n^2+1).
%F A245054 a(n) = A245049(n,n+1).
%F A245054 a(n) ~ 2^(n-1/2) * exp(n+1/4) * n^(n-5/2) / sqrt(Pi). - _Vaclav Kotesovec_, Jul 11 2014
%p A245054 a:= n-> add(binomial(n^2+i, i)*binomial(n^2+i+1, n-i), i=0..n)/(n^2+1):
%p A245054 seq(a(n), n=0..20);
%t A245054 Table[Sum[Binomial[n^2+i,i]*Binomial[n^2+i+1, n-i], {i,0,n}]/(n^2+1),{n,0,20}] (* _Vaclav Kotesovec_, Jul 11 2014 *)
%Y A245054 Main diagonal of A245049.
%K A245054 nonn
%O A245054 0,2
%A A245054 _Alois P. Heinz_, Jul 10 2014