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A135308
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Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k DUUU's.
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0
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1, 1, 2, 5, 13, 1, 35, 7, 96, 36, 267, 159, 3, 750, 645, 35, 2123, 2475, 264, 6046, 9136, 1602, 12, 17303, 32773, 8515, 195, 49721, 115017, 41349, 1925, 143365, 396730, 188010, 14740, 55, 414584, 1349440, 813072, 96200, 1144, 1201917, 4537368
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row n has floor((n+2)/3) terms (n>=1). Row sums yield the Cataln numbers (A000108). Coilumn 0 yields A005773. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 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)=(1/n)binom(n, k)Sum[(-1)^(j-k+1)*3^(n-j)*binom(n-k, j-k)binom(2j-2-3k, j-1),j=3k+1..n) (n>=1). G.f.F=F(t,z) satisfies tzF^3 + [3(1-t)z-1]F^2 - [3(1-t)z-1]F + (1-t)z = 0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007
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EXAMPLE
| Triangle begins:
1
1
2
5
13 1
35 7
96 36
267 159 3
...
T(5,1)=7 because we have UDUUUUDDDD, UDUUUDUDD, UDUUUDDUDD, UDUUUDDDUD, UDUDUUUDDD, UUDUUUDDDD and UUDDUUUDDD.
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MAPLE
| T:=proc(n, k) options operator, arrow: binomial(n, k)*(sum((-1)^(j-k+1)*3^(n-j)*binomial(n-k, j-k)*binomial(2*j-2-3*k, j-1), j=3*k+1..n))/n end proc: 1; for n to 15 do seq(T(n, k), k=0..floor((n-1)*1/3)) end do; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007
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CROSSREFS
| Cf. A000108, A005773.
Sequence in context: A042911 A137918 A114502 * A114492 A135305 A114463
Adjacent sequences: A135305 A135306 A135307 * A135309 A135310 A135311
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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 13 2007
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