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 A190167 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,... . 4
 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, 48, 52, 36, 18, 7, 2, 1, 38, 96, 115, 90, 51, 22, 8, 2, 1, 70, 193, 254, 217, 138, 68, 26, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS Row n has n-1 entries (n>=3). Sum of entries in row n is A004148(n) (the RNA secondary structure numbers). T(n,0)=A190168(n). Sum(kT(n,k),k>=0)=A190169(n). The trivariate g.f. H(t,s,z), where t (s) marks (1,0)-steps at even (odd) levels and z marks length, satisfies z^2(1-tz+z^2)H^2-(1-tz+z^2)(1-sz+z^2)H+1-sz+z^2=0. LINKS FORMULA 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. EXAMPLE 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). Triangle starts: 1; 1; 1; 1,1; 1,2,1; 2,3,2,1; 4,6,4,2,1; 7,12,10,5,2,1; MAPLE 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 CROSSREFS Cf. A004148, A190164, A190168, A190169 Sequence in context: A048685 A101050 A128979 * A171565 A328266 A115116 Adjacent sequences:  A190164 A190165 A190166 * A190168 A190169 A190170 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, May 06 2011 STATUS approved

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Last modified January 26 05:10 EST 2020. Contains 331273 sequences. (Running on oeis4.)