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A171850
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Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n for which the area below the path minus the number of U-steps is k (n>=0, k>=0).
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1
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1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 4, 4, 2, 3, 2, 1, 1, 5, 7, 5, 5, 5, 5, 2, 1, 1, 1, 6, 11, 10, 10, 10, 10, 8, 6, 4, 3, 2, 1, 1, 7, 16, 18, 18, 21, 21, 17, 16, 14, 11, 9, 7, 5, 2, 1, 1, 1, 8, 22, 30, 32, 38, 43, 40, 34, 32, 32, 26, 23, 20, 14, 10, 8, 4, 3, 2, 1, 1, 9, 29, 47, 55, 67, 79, 83
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OFFSET
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0,7
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COMMENTS
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The considered statistic (area below the path minus number of U-steps) in RNA secondary structure terminology is called density (see the Willenbring reference, p. 1611).
Number of entries in row n is 1 + floor((n-1)^2/4).
Sum of entries in row n = A004148(n) (the secondary structure numbers).
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LINKS
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FORMULA
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The trivariate g.f. G=G(t,u,z), where z marks length, t marks the area below the path, and x marks number of U-steps, satisfies G(t,x,z) = 1 + zG(t,x,z) + txz^2*(G(t,x,tz) - 1)G(t,x,z) (yielding a continued fraction expression for G(t,1/t,z)).
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EXAMPLE
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T(4,2)=1 because we have UHHD, where U=(1,1), H=(1,0), D=(1,-1).
Triangle starts:
1;
1;
1;
1, 1;
1, 2, 1;
1, 3, 2, 1, 1;
1, 4, 4, 2, 3, 2, 1;
1, 5, 7, 5, 5, 5, 5, 2, 1, 1;
1, 6, 11, 10, 10, 10, 10, 8, 6, 4, 3, 2, 1;
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MAPLE
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g[0] := 1/(1-z+z^2-z^2*g[1]): for n to 12 do g[n] := subs({z = t*z, g[n] = g[n+1]}, g[n-1]) end do: G := subs(g[16] = 0, g[0]): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 11 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*(n-1)^2)) end do; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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