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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
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OFFSET
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1,5
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COMMENTS
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Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
Sum_{k=1..n} k*T(n,k) = A162983(n).
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LINKS
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FORMULA
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G.f.: 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
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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
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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-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
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KEYWORD
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AUTHOR
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STATUS
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approved
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