0,6

The statistic "number of edges from the root to the first branchpoint" is equal to 0 if root is a branchpoint and it is equal to the total number of edges if there is no branchpoint. Row n contains n+1 terms. Row sums yield the ternary numbers (A001764). Column 0 yields A102893. Column 1 yields A030983.

M. Noy, Enumeration of noncrossing trees on a circle, Discr. Math. 180 (1998), 301-313.

T(n, 0)=5binomial(3n-1, n-2)/(3n-1) for n>0; T(n, k)=[2^(k-1)/(n-k+1)]binomial(3n-3k+1, n-k)-[2^k/(n-k)]binomial(3n-3k-2, n-k-1) for 0<k<n; T(n, n)=2^(n-1) (n>0). G.f.=(1/2)g(2-g)+g^2*(1-2z)/[2(1-2tz)], where g=1+zg^3 is the g.f. of the ternary numbers (A001764).

T(2,0)=1 because we have /\ and T(2,2)=2 because we have /_ and _\.

Triangle starts:

1;

0,1;

1,0,2;

5,3,0,4;

25,16,6,0,8;

130,83,32,12,0,16;

T:=proc(n, k) if n=0 and k=0 then 1 elif k=0 then 5*binomial(3*n-1, n-2)/(3*n-1) elif k<n then (2^(k-1)/(n-k+1))*binomial(3*n-3*k+1, n-k)-(2^k/(n-k))*binomial(3*n-3*k-2, n-k-1) elif k=n then 2^(n-1) else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

Cf. A001764, A102893, A030983.

Adjacent sequences: A102889 A102890 A102891 this_sequence A102893 A102894 A102895

Sequence in context: A108429 A130280 A011035 this_sequence A132898 A062706 A059217

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 16 2005