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%I #11 Nov 15 2019 21:02:42
%S 1,0,1,0,0,1,1,0,0,1,1,2,0,0,1,1,3,3,0,0,1,2,4,6,4,0,0,1,4,8,9,10,5,0,
%T 0,1,7,18,19,16,15,6,0,0,1,12,35,48,36,25,21,7,0,0,1,22,66,102,100,60,
%U 36,28,8,0,0,1,41,132,209,229,180,92,49,36,9,0,0,1,76,266,450,504,440,294,133,64,45,10,0,0,1
%N Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n having a total of k (1,0)-steps at levels 0,2,4,... .
%C Sum of entries in row n is A004148(n) (the RNA secondary structure numbers).
%C T(n,0)=A190165(n).
%C Sum_{k>=0} k*T(n,k) = A190166(n).
%C The trivariate g.f. H(t,s,z), where t (s) marks (1,0)-steps at even (odd) levels and z marks length, satisfies the equation
%C z^2*(1-tz+z^2)*H^2 - (1-tz+z^2)*(1-sz+z^2)*H + 1-sz+z^2 = 0.
%F G.f.: G = G(t,z) satisfies the equation z^2*(1-tz+z^2)*G^2 - (1-z+z^2)*(1-tz+z^2)*G + 1 - z + z^2 = 0.
%e T(5,2)=3 because we have h'h'uhd, h'uhdh', and uhdh'h', where u=(1,1), h=(1,0), d=(1,-1) (the even-level h-steps are marked).
%e Triangle starts:
%e 1;
%e 0, 1;
%e 0, 0, 1;
%e 1, 0, 0, 1;
%e 1, 2, 0, 0, 1;
%e 1, 3, 3, 0, 0, 1;
%p eq := z^2*(1-t*z+z^2)*G^2-(1-z+z^2)*(1-t*z+z^2)*G+1-z+z^2 = 0: g := RootOf(eq, G): Gser := simplify(series(g, z = 0, 15)): for n from 0 to 13 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
%t m = 13; G[_] = 0;
%t Do[G[z_] = -((z^2 G[z]^2 (-t z + z^2 + 1) + z^2 - z + 1)/((z^2 - z + 1)(t z - z^2 - 1))) + O[z]^m, {m}];
%t CoefficientList[#, t]& /@ CoefficientList[G[z], z] // Flatten (* _Jean-François Alcover_, Nov 15 2019 *)
%Y Cf. A004148, A190165, A190166, A110236, A190167.
%K nonn,tabl
%O 0,12
%A _Emeric Deutsch_, May 06 2011
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