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A135309
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k UUUU's starting at level 0.
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1
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1, 1, 2, 5, 13, 1, 36, 6, 105, 27, 319, 110, 1002, 427, 1, 3235, 1616, 11, 10685, 6034, 77, 35970, 22376, 440, 123045, 82725, 2241, 1, 426667, 305606, 10611, 16, 1496782, 1129683, 47823, 152, 5303623, 4181954, 208148, 1120
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row n has 1+floor(n/4) terms. Row sums yield the Catalan numbers (A000108). Column 0 yields A135310. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 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
| T(n,k)=Sum[(-1)^(j-k)*(5j+1)*binom(j,k)binom(2n-3j,n+j)/(n+j+1),j=k..floor(n/4)]. G.f.=G(t,z)=C/[1+(1-t)z^4*C^5], where C=[1-sqrt(1-4z)]/(2z) is the g.f. of the Catalan numbers (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 2007
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EXAMPLE
| Triangle begins:
1
1
2
5
13 1
36 6
105 27
319 110
1002 427 1
3235 1616 11
10685 6034 77
...
T(5,1)=6 because we have UUUUUDDDDD, UUUUDUDDDD, UUUUDDUDDD, UUUUDDDUDD, UUUUDDDUDD and UUUUDDDDUD.
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MAPLE
| T:=proc(n, k) options operator, arrow: sum((-1)^(j-k)*(5*j+1)*binomial(j, k)*binomial(2*n-3*j, n+j)/(n+j+1), j=k..floor((1/4)*n)) end proc: for n from 0 to 15 do seq(T(n, k), k=0..floor((1/4)*n)) end do; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 2007
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CROSSREFS
| Cf. A000108, A135310.
Sequence in context: A114492 A135305 A114463 * A135331 A135329 A114508
Adjacent sequences: A135306 A135307 A135308 * A135310 A135311 A135312
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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
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 2007
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