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A190172
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Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n having k UHD's; here U=(1,1), H=(1,0), and D=(1,-1).
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2
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1, 1, 1, 1, 1, 2, 2, 4, 4, 8, 8, 1, 16, 18, 3, 33, 40, 9, 69, 90, 25, 1, 146, 204, 69, 4, 312, 467, 183, 16, 673, 1074, 479, 56, 1, 1463, 2481, 1239, 185, 5, 3202, 5752, 3180, 576, 25, 7050, 13378, 8104, 1734, 105, 1, 15605, 31196, 20544, 5076, 405, 6, 34705, 72912, 51852, 14546, 1451, 36
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OFFSET
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0,6
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COMMENTS
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Number of entries in row n is 1+floor(n/3).
Sum of entries in row n = A004148 (the RNA secondary structure numbers).
Sum(k*T(n,k),k>=0)=A110236(n-2) (n>=3).
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LINKS
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FORMULA
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G.f. G=G(t,z) satisfies the equation G = 1 + zG + z^2*G(G-1-z+tz).
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EXAMPLE
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T(5,1)=4 because we have HHUHD, HUHDH, UHDH, and UUHDD.
Triangle starts:
1;
1;
1;
1,1;
2,2;
4,4;
8,8,1;
16,18,3;
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MAPLE
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eq := G = 1+z*G+z^2*G*(G-1-z+t*z): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) 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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