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A127535 Triangle read by rows: T(n,k) is the number of even trees with 2n edges and jump-length equal to k (0<=k<=n-1).

%I A127535 #8 Mar 22 2022 05:54:44
%S A127535 1,2,1,4,6,2,8,22,20,5,16,66,107,70,14,32,178,428,496,252,42,64,450,
%T A127535 1449,2498,2235,924,132,128,1090,4410,10234,13662,9878,3432,429,256,
%U A127535 2562,12479,36558,66107,71370,43043,12870,1430,512,5890,33512,118588
%N A127535 Triangle read by rows: T(n,k) is the number of even trees with 2n edges and jump-length equal to k (0<=k<=n-1).
%C A127535 An even tree is an ordered tree in which each vertex has an even outdegree. In the preorder traversal of an ordered tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump; the positive difference of the levels is called the jump distance; the sum of the jump distances in a given ordered tree is called the jump-length.
%C A127535 The Krandick reference considers jumps and jump-length only in full binary trees.
%H A127535 W. Krandick, <a href="https://doi.org/10.1016/j.cam.2003.08.018">Trees and jumps and real roots</a>, J. Computational and Applied Math., 162, 2004, 51-55.
%F A127535 G.f.: G=G(t,z) is given by (2t-1-t^2+2z-tz)G^3-(2+2tz-2t-5z)G^2+(4z-tz-1)G+z=0.
%F A127535 Sum of terms in row n = C(3n,n)/(2n+1) (A001764).
%F A127535 T(n,0)=2^(n-1) (A000079).
%F A127535 T(n+1,n)=C(2n,n)/(n+1) (A000108, the Catalan numbers).
%F A127535 Sum(k*T(n,k),0<=k<=n-1)=A127536(n).
%e A127535 Triangle starts:
%e A127535 1;
%e A127535 2,1;
%e A127535 4,6,2;
%e A127535 8,22,20,5;
%e A127535 16,66,107,70,14;
%p A127535 eq:=(2*t-1-t^2+2*z-t*z)*G^3-(2+2*t*z-2*t-5*z)*G^2+(4*z-t*z-1)*G+z: g:=RootOf(eq,G): gser:=simplify(series(g,z=0,14)): for n from 1 to 11 do P[n]:=sort(expand(coeff(gser,z,n))) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form
%Y A127535 Cf. A001764, A000079, A000108, A127536, A127529, A127532.
%K A127535 nonn,tabl
%O A127535 1,2
%A A127535 _Emeric Deutsch_, Jan 19 2007