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A190172 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).

%I A190172 #4 Mar 30 2012 17:36:27
%S A190172 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 A190172 467,183,16,673,1074,479,56,1,1463,2481,1239,185,5,3202,5752,3180,576,
%U A190172 25,7050,13378,8104,1734,105,1,15605,31196,20544,5076,405,6,34705,72912,51852,14546,1451,36
%N A190172 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 A190172 Number of entries in row n is 1+floor(n/3).
%C A190172 Sum of entries in row n = A004148 (the RNA secondary structure numbers).
%C A190172 T(n,0)=A004149(n).
%C A190172 Sum(k*T(n,k),k>=0)=A110236(n-2) (n>=3).
%F A190172 G.f. G=G(t,z) satisfies the equation G = 1 + zG + z^2*G(G-1-z+tz).
%e A190172 T(5,1)=4 because we have HHUHD, HUHDH, UHDH, and UUHDD.
%e A190172 Triangle starts:
%e A190172 1;
%e A190172 1;
%e A190172 1;
%e A190172 1,1;
%e A190172 2,2;
%e A190172 4,4;
%e A190172 8,8,1;
%e A190172 16,18,3;
%p A190172 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 A190172 Cf. A004148, A004149, A110236
%K A190172 nonn,tabf
%O A190172 0,6
%A A190172 _Emeric Deutsch_, May 06 2011