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A127155
Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k long ascents and long descents. A long ascent (descent) in a Dyck path is a maximal sequence of at least 2 consecutive up (down) steps.
0
1, 1, 1, 0, 1, 1, 0, 4, 1, 0, 10, 2, 1, 1, 0, 20, 12, 9, 1, 0, 35, 42, 47, 6, 1, 1, 0, 56, 112, 180, 64, 16, 1, 0, 84, 252, 558, 374, 148, 12, 1, 1, 0, 120, 504, 1482, 1580, 950, 200, 25, 1, 0, 165, 924, 3498, 5390, 4662, 1770, 365, 20, 1, 1, 0, 220, 1584, 7524, 15752, 18676
OFFSET
0,8
COMMENTS
Row n has 1+2*floor(n/2) terms. Row sums yield the Catalan numbers (A000108).
FORMULA
G.f.: G=G(t,z) satisfies: z(1-z+tz)^2*G^2-(1-z+tz)(1+z-tz)G+1 = 0.
EXAMPLE
T(4,3)=2 because we have (UU)D(UU)(DDD) and (UUU)(DD)U(DD) (here U=(1,1) and D=(1,-1); the long ascents and the long descents are shown between parentheses).
Triangle starts:
1;
1;
1,0,1;
1,0,4;
1,0,10,2,1;
1,0,20,12,9;
1,0,35,42,47,6,1;
MAPLE
G:=(1+z-t*z-sqrt(t^2*z^2-2*t*z^2-2*t*z+z^2-2*z+1))/2/z/(1-z+t*z): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 12 do seq(coeff(P[n], t, j), j=0..2*floor(n/2)) od; # yields sequence in triangular form
CROSSREFS
Cf. A000108.
Sequence in context: A284982 A089962 A363971 * A211793 A145880 A048516
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Feb 27 2007
STATUS
approved