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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(2n-1,1) = A000957(n) (the Fine numbers).
LINKS
FORMULA
G.f.: 2(1 + (t-1)z(1-2z) + q(1 - z + tz))/((1-2z+q)(1+2z^2-2t^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(1-4*z^2)-2*z^2-z*t-1-sqrt(1-4*z^2)+2*z^2*t-z*t*sqrt(1-4*z^2))/(-1-sqrt(1-4*z^2)+2*z)/(-1-sqrt(1-4*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
Sequence in context: A091866 A168511 A111146 * A355916 A309023 A353429
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jun 17 2005
STATUS
approved

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Last modified July 13 09:13 EDT 2024. Contains 374274 sequences. (Running on oeis4.)