0,8

Also number of ordered trees with n edges and having k leaves that are endpoints of branches of length 1 (i.e. leaf is child of the root or of a branchnode). Row sums are the Catalan numbers (A000108). Column 0 yields A004148. Sum(kT(n,k),k=0..n)=A097613(n).

G.f.=G=G(t, z) satisfies G=1+zG(G-1+t-tz+z).

T(4,2)=4 because we have uu(ud)(ud)dd, uudd(ud)(ud), (ud)uudd(ud) and

(ud)(ud)uudd (the peaks that are not of the form uudd are shown between parentheses).

Triangle begins:

1;

0,1;

1,0,1;

1,2,1,1;

2,4,4,3,1;

4,9,13,9,6,1

G:=1/2/z*(1-z^2+z-t*z+z^2*t-sqrt(1-z^2+z^4-2*z^3+4*z^3*t-2*z^4*t+t^2*z^2-2*t^2*z^3+z^4*t^2-2*t*z-2*z)): Gser:=simplify(series(G, z=0, 15)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 13 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form

Cf. A000108, A004148, A097613.

Sequence in context: A386987 A253473 A026120 * A333270 A333272 A352501

Adjacent sequences: A108743 A108744 A108745 * A108747 A108748 A108749

nonn,tabl

Emeric Deutsch, Jun 23 2005

approved