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A114502 Triangle read by rows: T(n,k) is number of ordered trees with n edges and having exactly k vertices all of whose children are leaves (1<=k<=floor(n/2) for n>=2).

%I A114502 #13 Apr 11 2024 10:50:00
%S A114502 1,2,5,13,1,34,8,89,42,1,233,183,13,610,717,102,1,1597,2622,624,19,
%T A114502 4181,9134,3275,205,1,10946,30691,15473,1650,26,28657,100284,67684,
%U A114502 11020,366,1,75025,320466,279106,64553,3716,34,196418,1005630,1098402,342867
%N A114502 Triangle read by rows: T(n,k) is number of ordered trees with n edges and having exactly k vertices all of whose children are leaves (1<=k<=floor(n/2) for n>=2).
%C A114502 Row 1 has one term; row n (n>=2) has floor(n/2) terms. Row sums are the Catalan numbers (A000108). Column 1 yields the Fibonacci numbers with odd index (A001519). Sum(kT(n,k),k=1..floor(n/2))=[1+sum(binomial(2j,j),j=0..n-1)]/2 (A024718).
%H A114502 Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, <a href="https://arxiv.org/abs/2404.05672">Enumerating runs, valleys, and peaks in Catalan words</a>, arXiv:2404.05672 [math.CO], 2024. See p. 14.
%F A114502 G.f.: G=G(t, z) satisfies z(1-z)G^2-(1-z-z^2+tz^2)G+1-2z+tz=0.
%F A114502 G.f. G(t,z) can be derived easily from the symbolic decomposition of an ordered tree according to the degree of the root; one obtains G = 1 + z*(G-1+t) + z^2*(G^2-1+t) + z^3*(G^3-1+t) + ... . - _Emeric Deutsch_, Feb 12 2015
%e A114502 T(4,2)=1 because we have the tree with two paths of length two, rab and rcd, emanating from the root r; a and b are vertices all of whose children are leaves.
%e A114502 Triangle starts:
%e A114502   1;
%e A114502   2;
%e A114502   5;
%e A114502   13,1;
%e A114502   34,8;
%e A114502   89,42,1;
%e A114502   233,183,13;
%e A114502   610,717,102,1;
%e A114502   ...
%p A114502 G:=1/2/(z^2-z)*(-1+z+z^2-t*z^2+sqrt(1-6*z+11*z^2-2*t*z^2-6*z^3+2*z^3*t+z^4-2*z^4*t+t^2*z^4)): Gser:=simplify(series(G,z=0,18)): for n from 1 to 15 do P[n]:=coeff(Gser,z^n) od: 1; for n from 1 to 15 do seq(coeff(P[n],t^j),j=1..floor(n/2)) od; # yields sequence in triangular form
%Y A114502 Cf. A000108, A001519, A024718.
%K A114502 nonn,tabf
%O A114502 1,2
%A A114502 _Emeric Deutsch_, Dec 02 2005