0,4

Column k (k>=1) starts with 0, followed by the partial sums of column k-1. Row sums yield A126221.

Indexing n and k from 1 instead of from 0, T(n,k) is the number of Dyck n-paths whose first peak is at height k and whose first component avoids DUU. A primitive Dyck path is one whose only return (to ground level) is at the end. The interior returns of a general Dyck path split the path into a list of primitive Dyck paths, called its components. For example, UUDDUD has components UUDD, UD and T(4,2) = 4 counts UUDUDUDD, UUDDUUDD, UUDDUDUD, UUDUDDUD (but not UUDUUDDD because its first component contains a DUU). - David Callan (callan(AT)stat.wisc.edu), Jan 17 2007

G.f.=G(t,x)=(1-x)[1-sqrt(1-4x)]/[2x(1-x-tx)].

First few rows of the triangle are:

1;

1, 1;

2, 2, 1;

5, 4, 3, 1;

14, 9, 7, 4, 1;

42, 23, 16, 11, 5, 1;

...

(5,3) = 16 = 7 + 9 = (4,3) + (4,2).

T:=proc(n, k) if k=0 then binomial(2*n, n)/(n+1) elif n=0 then 0 else T(n-1, k)+T(n-1, k-1) fi end: for n from 0 to 11 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

G:=(1-x)*(1-sqrt(1-4*x))/2/x/(1-x-t*x): Gser:=simplify(series(G, x=0, 15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, x, n)) od: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form

Cf. A000108, A125178, A126221.

Sequence in context: A110438 A121460 A105292 this_sequence A125178 A101975 A136388

Adjacent sequences: A125174 A125175 A125176 this_sequence A125178 A125179 A125180

nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2006

Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 28 2006