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A162984
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Number of Dyck paths with no UUU's and no DDD's of semilength n and having k UUDUDD's (0<=k<=floor(n/3); U=(1,1), D=(1,-1)).
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2
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1, 1, 2, 3, 1, 6, 2, 12, 5, 25, 11, 1, 53, 26, 3, 114, 62, 9, 249, 148, 25, 1, 550, 355, 69, 4, 1227, 853, 189, 14, 2760, 2057, 509, 46, 1, 6253, 4973, 1359, 145, 5, 14256, 12050, 3600, 446, 20, 32682, 29256, 9484, 1334, 75, 1, 75293, 71154, 24870, 3914, 265, 6
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| T(n,k) is the number of weighted lattice paths in B(n) having k peaks. The members of B(n) are paths of weight n that start at (0,0), end on but never go below the horizontal axis, and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps. A peak is a (1,1)-step followed by a (1,-1)-step. Example: T(7,2)=3. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), we have hUDUD, UDhUD, and UDUDh.
Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
T(n,0)=A162985(n).
Sum(k*T(n,k), k=0..floor(n/3))=A110320(n-2).
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REFERENCES
| M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
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FORMULA
| G.f. G=G(t,z) satisfies G = 1 + zG + z^2*G + z^3*(G-1+t)G.
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EXAMPLE
| T(4,1)=2 because we have UDUUDUDD and UUDUDDUD.
Triangle starts:
1;
1;
2;
3,1;
6,2;
12,5;
25,11,1;
53,26,3;
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MAPLE
| G := ((1-z-z^2+z^3-t*z^3-sqrt(1-2*z-z^2-2*t*z^3-z^4-2*z^5+z^6+2*t*z^4+2*t*z^5-2*t*z^6+t^2*z^6))*1/2)/z^3: Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
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CROSSREFS
| A004148, A110320, A162985
Sequence in context: A156344 A119440 A165742 * A166295 A016730 A114576
Adjacent sequences: A162981 A162982 A162983 * A162985 A162986 A162987
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 11 2009
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