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Search: id:A127153
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| A127153 |
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having k UDUD's starting at level 0; here U=(1,1), D=(1,-1) (0<=k<=n-1). |
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+0
2
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| 1, 1, 1, 1, 4, 0, 1, 11, 2, 0, 1, 33, 6, 2, 0, 1, 105, 17, 7, 2, 0, 1, 343, 56, 19, 8, 2, 0, 1, 1148, 185, 64, 21, 9, 2, 0, 1, 3916, 624, 214, 72, 23, 10, 2, 0, 1, 13563, 2144, 726, 244, 80, 25, 11, 2, 0, 1, 47571, 7468, 2510, 832, 275, 88, 27, 12, 2, 0, 1, 168625, 26317
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OFFSET
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0,5
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COMMENT
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Row 0 has one entry; row n has n entries (n>=1). Row sums yield the Catalan numbers (A000108). Column 0 yields A135338. The reference does not list the 0's (p. 2920, lines 3,4).
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REFERENCES
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A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
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FORMULA
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G.f.=(1+z-tz)/[1+z-tz+z^2-tz^2-zC(1+z-tz)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan g.f. (see A000108).
Or, g.f.=[1+(1-t)z]C/[1+(1-t)z(1+zC)].
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EXAMPLE
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T(4,1)=2 because we have UDUDUUDD and UUDDUDUD; T(4,3)=1 because we have UDUDUDUD.
Triangle starts:
1;
1;
1,1;
4,0,1;
11,2,0,1;
33,6,2,0,1;
105,17,7,2,0,1;
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MAPLE
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G:=(1+z-t*z)/(1+z-t*z+z^2-t*z^2-z*C*(1+z-t*z)): C:=(1-sqrt(1-4*z))/2/z: Gser:=simplify(series(G, z=0, 16)): for n from 0 to 13 do P[n]:=sort(coeff(Gser, z, n)) od: 1; for n from 1 to 13 do seq(coeff(P[n], t, j), j=0..n-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000108, A127154, A135338.
Sequence in context: A121408 A121301 A059056 this_sequence A054375 A136452 A067565
Adjacent sequences: A127150 A127151 A127152 this_sequence A127154 A127155 A127156
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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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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), May 16 2008 at the suggestion of R. J. Mathar.
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