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Search: id:A135329
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| A135329 |
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k DDUU's starting at level 2. |
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+0
2
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| 1, 1, 2, 5, 13, 1, 36, 6, 106, 25, 1, 327, 94, 8, 1045, 342, 42, 1, 3433, 1230, 189, 10, 11529, 4411, 792, 63, 1, 39414, 15840, 3192, 328, 12, 136733, 57074, 12569, 1547, 88, 1, 480180, 206522, 48787, 6878, 519, 14, 1703807, 750693, 187680, 29427, 2715, 117
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Each of the rows 0,1,2,3 contains one entry. Row n (n>=4) contain floor(n/2) entries. Column 0 is A135335. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007
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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+(1-t)z^2*C^2]C/[1+(1-t)(1-z)z^2*C^3], where C=[1-sqrt(1-4z)]/(2z) is the g.f. of the Catalan numbers (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007
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EXAMPLE
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Triangle begins:
1
1
2
5
13 1
36 6
106 25 1
327 94 8
1045 342 42 1
3433 1230 189 10
...
T(5,1)=6 because we have UDUU(DDUU)DD, UU(DDUU)DDUD, UU(DDUU)DUDD, UU(DDUU)UDDD, UUDU(DDUU)DD, and UUUD(DDUU)DD (the DDUU's starting at level 2 are shown between parentheses).
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MAPLE
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G:=C*(1+(1-t)*z^2*C^2)/(1+(1-t)*(1-z)*z^2*C^3): C:=((1-sqrt(1-4*z))*1/2)/z: Gser:=simplify(series(G, z=0, 20)): for n from 0 to 16 do P[n]:=sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 16 do seq(coeff(P[n], t, j), j=0..floor((n-2)*1/2)) end do; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007
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CROSSREFS
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Cf. A000108, A135335.
Adjacent sequences: A135326 A135327 A135328 this_sequence A135330 A135331 A135332
Sequence in context: A114463 A135309 A135331 this_sequence A114508 A139023 A067365
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KEYWORD
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nonn,tabf
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AUTHOR
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njas, Dec 07 2007
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EXTENSIONS
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Edited and extended by Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007
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