

A109077


Triangle read by rows: T(n,k) is the number of symmetric Dyck paths of semilength n and having k hills (i.e., peaks at level 1).


1



1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 4, 0, 1, 0, 1, 6, 1, 2, 0, 0, 1, 13, 0, 5, 0, 1, 0, 1, 22, 2, 6, 2, 2, 0, 0, 1, 46, 0, 16, 0, 6, 0, 1, 0, 1, 80, 6, 24, 4, 6, 3, 2, 0, 0, 1, 166, 0, 58, 0, 19, 0, 7, 0, 1, 0, 1, 296, 18, 90, 13, 26, 6, 6, 4, 2, 0, 0, 1, 610, 0, 211, 0, 71, 0, 22, 0, 8, 0, 1, 0, 1, 1106
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,7


COMMENTS

Column 0 yields A109078.
T(2n,1)=0, T(2n1,1) = A000957(n) (the Fine numbers).


LINKS

Table of n, a(n) for n=0..91.


FORMULA

G.f.: 2(1 + (t1)z(12z) + q(1  z + tz))/((12z+q)(1+2z^22t^2*z^2+q)), where q = sqrt(1  4z^2).


EXAMPLE

T(5,2)=2 because we have uduududdud and uduuudddud, where u=(1,1), d=(1,1).
Triangle begins:
1;
0, 1;
1, 0, 1;
2, 0, 0, 1;
4, 0, 1, 0, 1;
6, 1, 2, 0, 0, 1;


MAPLE

G:=2*(z+z*sqrt(14*z^2)2*z^2z*t1sqrt(14*z^2)+2*z^2*tz*t*sqrt(14*z^2))/(1sqrt(14*z^2)+2*z)/(1sqrt(14*z^2)2*z^2+2*z^2*t^2): Gser:=simplify(series(G, z=0, 17)): 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


CROSSREFS

Cf. A000957, A109078.
Sequence in context: A091866 A168511 A111146 * A137585 A301344 A301579
Adjacent sequences: A109074 A109075 A109076 * A109078 A109079 A109080


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Jun 17 2005


STATUS

approved



