A127538 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).

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, 101, 87, 12, 16, 0, 1, 134, 503, 448, 177, 133, 15, 19, 0, 1, 411, 1784, 1305, 862, 271, 188, 18, 22, 0, 1, 1411, 5887, 4848, 2524, 1444, 383, 252, 21, 25, 0, 1, 4747, 20604

Offset

0,8

Comments

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

Links

Table of n, a(n) for n=0..67.

Formula

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

Example

T(2,2)=1 because we have the tree /\.
Triangle starts:
1;
0,1;
1,0,1;
0,4,0,1;
3,3,7,0,1;
3,22,6,10,0,1;

Maple

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

Crossrefs

Cf. A000108, A127539, A127540, A127541.

Sequence in context: A155998 A298063 A298712 * A096008 A122873 A221275

Adjacent sequences: A127535 A127536 A127537 * A127539 A127540 A127541

Keyword

nonn,tabl

Author

Emeric Deutsch, Mar 01 2007

Status

approved