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Search: id:A162986
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| A162986 |
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Number of Dyck paths with no UUU's and no DDD's of semilength n and having k UD's starting at level 0 (i.e. hills); (0<=k<=n; U=(1,1), D=(1,-1)). |
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+0
1
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| 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 2, 3, 0, 1, 4, 5, 3, 4, 0, 1, 8, 10, 9, 4, 5, 0, 1, 17, 21, 18, 14, 5, 6, 0, 1, 37, 46, 40, 28, 20, 6, 7, 0, 1, 82, 102, 90, 66, 40, 27, 7, 8, 0, 1, 185, 230, 204, 152, 100, 54, 35, 8, 9, 0, 1, 423, 526, 469, 353, 235, 143, 70, 44, 9, 10, 0, 1, 978, 1216
(list; table; graph; listen)
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OFFSET
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0,8
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COMMENT
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Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
T(n,0)=A004148(n-1) (n>=1).
Sum(k*T(n,k), k=0..n)=A162987(n).
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FORMULA
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G(t,z)=1/(1-tz-z^2-z^3*g) , where g=1+zg+z^2*g+z^3*g^2.
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EXAMPLE
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T(5,2)=3 because we have (UD)(UD)UUDUDD, (UD)UUDUDD(UD), and UUDUDD(UD)(UD) (the hills are placed between parentheses).
Triangle starts:
1;
0,1;
1,0,1;
1,2,0,1;
2,2,3,0,1;
4,5,3,4,0,1
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MAPLE
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g := ((1-z-z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^3: G := 1/(1-t*z-z^2-z^3*g): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form
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CROSSREFS
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A004148, A162987
Sequence in context: A128627 A105422 A166291 this_sequence A128584 A080099 A127711
Adjacent sequences: A162983 A162984 A162985 this_sequence A162987 A162988 A162989
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 11 2009
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