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%I #4 Mar 30 2012 17:36:27
%S 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,
%T 467,183,16,673,1074,479,56,1,1463,2481,1239,185,5,3202,5752,3180,576,
%U 25,7050,13378,8104,1734,105,1,15605,31196,20544,5076,405,6,34705,72912,51852,14546,1451,36
%N 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).
%C Number of entries in row n is 1+floor(n/3).
%C Sum of entries in row n = A004148 (the RNA secondary structure numbers).
%C T(n,0)=A004149(n).
%C Sum(k*T(n,k),k>=0)=A110236(n-2) (n>=3).
%F G.f. G=G(t,z) satisfies the equation G = 1 + zG + z^2*G(G-1-z+tz).
%e T(5,1)=4 because we have HHUHD, HUHDH, UHDH, and UUHDD.
%e Triangle starts:
%e 1;
%e 1;
%e 1;
%e 1,1;
%e 2,2;
%e 4,4;
%e 8,8,1;
%e 16,18,3;
%p 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
%Y Cf. A004148, A004149, A110236
%K nonn,tabf
%O 0,6
%A _Emeric Deutsch_, May 06 2011
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