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%I #4 Mar 30 2012 17:36:27
%S 1,1,1,1,1,1,2,1,2,3,2,1,4,6,4,2,1,7,12,10,5,2,1,12,24,23,14,6,2,1,21,
%T 48,52,36,18,7,2,1,38,96,115,90,51,22,8,2,1,70,193,254,217,138,68,26,
%U 9,2,1,130,388,559,522,358,196,87,30,10,2,1,243,782,1220,1240,926,542,264,108,34,11,2,1
%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 1,3,5,... .
%C Row n has n-1 entries (n>=3).
%C Sum of entries in row n is A004148(n) (the RNA secondary structure numbers).
%C T(n,0)=A190168(n).
%C Sum(kT(n,k),k>=0)=A190169(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
%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(s,z) satisfies the equation z^2*(1-z+z^2)G^2-(1-z+z^2)(1-sz+z^2)G+1-sz+z^2=0.
%e T(5,2)=2 because we have huh'h'd and uh'h'dh, where u=(1,1), h=(1,0), d=(1,-1) (the odd-level h-steps are marked).
%e Triangle starts:
%e 1;
%e 1;
%e 1;
%e 1,1;
%e 1,2,1;
%e 2,3,2,1;
%e 4,6,4,2,1;
%e 7,12,10,5,2,1;
%p eq:=z^2*(1-z+z^2)*G^2-(1-z+z^2)*(1-s*z+z^2)*G+1-s*z+z^2 = 0: g:= RootOf(eq, G): Gser:= simplify(series(g, z = 0, 17)): for n from 0 to 13 do P[n] := sort(expand(coeff(Gser, z, n))) end do: 1; 1; for n from 0 to 13 do seq(coeff(P[n], s, k), k = 0 .. n-2) end do; # yields sequence in triangular form
%Y Cf. A004148, A190164, A190168, A190169
%K nonn,tabf
%O 0,7
%A _Emeric Deutsch_, May 06 2011
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