A127538 - OEIS

%I #2 Mar 30 2012 17:36:12

%S 1,0,1,1,0,1,0,4,0,1,3,3,7,0,1,3,22,6,10,0,1,16,43,50,9,13,0,1,37,175,

%T 101,87,12,16,0,1,134,503,448,177,133,15,19,0,1,411,1784,1305,862,271,

%U 188,18,22,0,1,1411,5887,4848,2524,1444,383,252,21,25,0,1,4747,20604

%N Triangle read by rows: T(n,k) is the number of ordered trees with n edges having k odd-length branches starting at the root (0<=k<=n).

%C Row sums are the Catalan numbers (A000108). T(n,0)=A127539(n). Sum(k*T(n,k),k=0..n)=A127540(n).

%F G.f.=(1+z)/(1+z-z^2*C-tzC), where C =[1-sqrt(1-4z)]/(2z) is the Catalan function.

%e T(2,2)=1 because we have the tree /\.

%e Triangle starts:

%e 1;

%e 0,1;

%e 1,0,1;

%e 0,4,0,1;

%e 3,3,7,0,1;

%e 3,22,6,10,0,1;

%p C:=(1-sqrt(1-4*z))/2/z: G:=(1+z)/(1+z-z^2*C-t*z*C): Gser:=simplify(series(G,z=0,15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

%Y Cf. A000108, A127539, A127540, A127541.

%K nonn,tabl

%O 0,8

%A _Emeric Deutsch_, Mar 01 2007