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A135305
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Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k UUUU's.
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0
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1, 1, 2, 5, 13, 1, 36, 5, 1, 104, 21, 6, 1, 309, 84, 28, 7, 1, 939, 322, 124, 36, 8, 1, 2905, 1206, 522, 174, 45, 9, 1, 9118, 4455, 2127, 795, 235, 55, 10, 1, 28964, 16302, 8492, 3487, 1155, 308, 66, 11, 1, 92940, 59268, 33396, 14894, 5412, 1617, 394, 78, 12, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Each of the rows 0,1,2,3 has one entry. Row n (n>=3) has n-2 entries. Row sums yield the Catalan numbers (A000108). Column 0 yields A036765. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007
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REFERENCES
| A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
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FORMULA
| G.f.=G=G(t,z) satisfies (1-t)z^3*G^3+z(t+z-tz)G^2+((1-t)z-1)G+1=0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007
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EXAMPLE
| Triangle begins:
1
1
2
5
13 1
36 5 1
104 21 6 1
309 84 28 7 1
...
T(5,1)=5 because we have UUUUDUDDDD, UUUUDDUDDD, UUUUDDDUDD, UUUUDDDDUD and UDUUUUDDDD.
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MAPLE
| eq:=(1-t)*z^3*G^3+z*(t+z-t*z)*G^2+((1-t)*z-1)*G+1: g:=RootOf(eq, G): gser:= simplify(series(g, z=0, 15)): for n from 0 to 12 do P[n]:=sort(coeff(gser, z, n)) end do: 1; 1; 2; for n from 3 to 12 do seq(coeff(P[n], t, j), j=0..n-3) end do; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007
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CROSSREFS
| Cf. A000108, A036765.
Sequence in context: A114502 A135308 A114492 * A114463 A135309 A135331
Adjacent sequences: A135302 A135303 A135304 * A135306 A135307 A135308
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KEYWORD
| nonn,tabf
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 07 2007
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EXTENSIONS
| Edited and extended by Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007
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