0,7

Row sums are the RNA secondary structure numbers (A004148). Column 1 without the zeros yields A077855.

Alois P. Heinz, Rows n = 0..200, flattened

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.

P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Sem. Loth. Comb. B08l (1984) 79-86.

G.f.: 1+sum(t^j*z^(2j+1)/[P(j)*P(j+1)], j=0..infinity), where P(j) are polynomials in z defined by P(0)=1, P(1)=1-z, P(j)=(1-z+z^2)P(j-1) -z^2*P(j-2), j=2, 3, ... .

Triangle starts:

1;

1;

1;

1, 1;

1, 3;

1, 6, 1;

1, 11, 5;

1, 20, 15, 1;

1, 36, 38, 7;

1, 64, 91, 28, 1;

1, 113, 211, 89, 9;

1, 199, 477, 255, 45, 1;

Row n >0 has ceil(n/2) terms.

T(6,2) = 5 because the peakless Motzkin paths of length 6 and height 2 are HUUHDD, UHUHDD, UUHHDD, UUHDDH, UUHDHD, where U=(1,1), H=(1,0) and D=(1,-1).

P[0]:=1: P[1]:=sort(1-z): for j from 2 to 30 do P[j]:=sort(expand((1-z+z^2)*P[j-1]-z^2*P[j-2])) od: G:=1+sum(t^i*z^(2*i+1)/P[i]/P[i+1], i=0..25): Gser:=simplify(series(G, z=0, 21)): Q[0]:=1: for m from 1 to 18 do Q[m]:=sort(coeff(Gser, z^m)) od: 1, seq(seq(coeff(t*Q[n], t^k), k=1..ceil(n/2)), n=1..16);

max = 16; p[0] = 1; p[1] := 1 - z; p[j_] := p[j] = (1 - z + z^2)*p[j - 1] - z^2*p[j - 2]; gf = 1 + Sum[t^j*z^(2*j + 1)/(p[j]*p[j + 1]), {j, 0, max}]; se = Series[gf, {t, 0, max}, {z, 0, max}]; CoefficientList[se, {z, t}] // DeleteCases[#, 0, 2] & // Flatten (* Jean-François Alcover, Jun 25 2013 *)

Cf. A004148, A077855.

Sequence in context: A034839 A089732 A158905 * A171852 A010287 A130270

Adjacent sequences: A098073 A098074 A098075 * A098077 A098078 A098079

nonn,tabf

Emeric Deutsch, Sep 13 2004

approved