A127536 -id:A127536 - cp's OEIS frontend

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).

1, 2, 1, 4, 6, 2, 8, 22, 20, 5, 16, 66, 107, 70, 14, 32, 178, 428, 496, 252, 42, 64, 450, 1449, 2498, 2235, 924, 132, 128, 1090, 4410, 10234, 13662, 9878, 3432, 429, 256, 2562, 12479, 36558, 66107, 71370, 43043, 12870, 1430, 512, 5890, 33512, 118588

Offset: 1

Emeric Deutsch, Jan 19 2007

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.

The Krandick reference considers jumps and jump-length only in full binary trees.

			Triangle starts:
1;
2,1;
4,6,2;
8,22,20,5;
16,66,107,70,14;

W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 51-55.

Cf. A001764, A000079, A000108, A127536, A127529, A127532.

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

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.
Sum of terms in row n = C(3n,n)/(2n+1) (A001764).
T(n,0)=2^(n-1) (A000079).
T(n+1,n)=C(2n,n)/(n+1) (A000108, the Catalan numbers).
Sum(k*T(n,k),0<=k<=n-1)=A127536(n).

A127534 Number of jumps in all even trees with 2n edges.

Original entry on oeis.org

0, 1, 9, 65, 442, 2940, 19380, 127281, 834900, 5476185, 35937525, 236030652, 1551652424, 10210456360, 67254204696, 443410005585, 2926078447656, 19325957314755, 127746785056275, 845069382939705, 5594334252541650

Offset: 1

Emeric Deutsch, Jan 19 2007

nonn

The Krandick reference considers jumps in full binary trees.

W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 51-55.

Cf. A127535, A127536.

seq((n-1)*(4*n-3)*binomial(3*n,n)/3/(2*n+1)/(3*n-1),n=1..24);

Table[((n-1)(4n-3)Binomial[3n,n])/(3(2n+1)(3n-1)),{n,30}] (* Harvey P. Dale, Sep 29 2013 *)

a(n)=(n-1)(4n-3)C(3n,n)/[3(2n+1)(3n-1)].

D-finite with recurrence 8*n*(2*n+1)*a(n) -2*(136*n-69)*(n-1)*a(n-1) +5*(263*n^2-893*n+750)*a(n-2) -156*(3*n-8)*(3*n-10)*a(n-3)=0. - R. J. Mathar, Jul 22 2022

cp's OEIS Frontend

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).

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