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Search: id:A127155
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| A127155 |
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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. |
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+0
1
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| 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
(list; graph; listen)
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OFFSET
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0,8
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COMMENT
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Row n has 1+2*floor(n/2) terms. Row sums yield the Catalan numbers (A000108).
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FORMULA
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G.f.=G=G(t,z) satisfies: z(1-z+tz)^2*G^2-(1-z+tz)(1+z-tz)G+1 = 0.
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EXAMPLE
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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;
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MAPLE
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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
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CROSSREFS
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Cf. A000108.
Sequence in context: A134575 A095831 A089962 this_sequence A048516 A060638 A007789
Adjacent sequences: A127152 A127153 A127154 this_sequence A127156 A127157 A127158
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2007
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