OFFSET
1,2
COMMENTS
LINKS
Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See p. 14.
FORMULA
G.f.: G=G(t, z) satisfies z(1-z)G^2-(1-z-z^2+tz^2)G+1-2z+tz=0.
G.f. G(t,z) can be derived easily from the symbolic decomposition of an ordered tree according to the degree of the root; one obtains G = 1 + z*(G-1+t) + z^2*(G^2-1+t) + z^3*(G^3-1+t) + ... . - Emeric Deutsch, Feb 12 2015
EXAMPLE
T(4,2)=1 because we have the tree with two paths of length two, rab and rcd, emanating from the root r; a and b are vertices all of whose children are leaves.
Triangle starts:
1;
2;
5;
13,1;
34,8;
89,42,1;
233,183,13;
610,717,102,1;
...
MAPLE
G:=1/2/(z^2-z)*(-1+z+z^2-t*z^2+sqrt(1-6*z+11*z^2-2*t*z^2-6*z^3+2*z^3*t+z^4-2*z^4*t+t^2*z^4)): Gser:=simplify(series(G, z=0, 18)): for n from 1 to 15 do P[n]:=coeff(Gser, z^n) od: 1; for n from 1 to 15 do seq(coeff(P[n], t^j), j=1..floor(n/2)) od; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Dec 02 2005
STATUS
approved