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A162981
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Number of Dyck paths with no UUU's and no DDD's of semilength n and having k returns to the x-axis (1<=k<=n; U=(1,1), D=(1,-1)).
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0
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1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 4, 6, 4, 1, 4, 7, 10, 10, 5, 1, 8, 14, 18, 20, 15, 6, 1, 17, 29, 36, 39, 35, 21, 7, 1, 37, 62, 76, 80, 75, 56, 28, 8, 1, 82, 136, 165, 172, 161, 132, 84, 36, 9, 1, 185, 304, 366, 380, 355, 300, 217, 120, 45, 10, 1, 423, 690, 826, 855, 800, 684, 525
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
T(n,1)=A004148(n-2)(n>=2).
Sum(k*T(n,k), k=1..n)=A162983(n).
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FORMULA
| G(t,z)=1/(1-tz-tz^2-tz^3*g) - 1, where g=1+zg+z^2*g+z^3*g^2.
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EXAMPLE
| T(5,2)=4 because we have UD'UUDUDUDD', UUDD'UUDUDD', UUDUDD'UUDD', and UUDUDUDD'UD' (the return steps are marked).
Triangle starts:
1;
1,1;
1,2,1;
1,3,3,1;
2,4,6,4,1;
4,7,10,10,5,1
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MAPLE
| 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-t*z^2-t*z^3*g)-1: Gser := simplify(series(G, z = 0, 16)): for n to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 12 do seq(coeff(P[n], t, j), j = 1 .. n) end do; # yields sequence in triangular form
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CROSSREFS
| A004148, A162983
Sequence in context: A170925 A030189 A114162 * A029264 A124054 A082870
Adjacent sequences: A162978 A162979 A162980 * A162982 A162983 A162984
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 11 2009
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