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).

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, 467, 183, 16, 673, 1074, 479, 56, 1, 1463, 2481, 1239, 185, 5, 3202, 5752, 3180, 576, 25, 7050, 13378, 8104, 1734, 105, 1, 15605, 31196, 20544, 5076, 405, 6, 34705, 72912, 51852, 14546, 1451, 36

Offset: 0

Author

Emeric Deutsch, May 06 2011

Keywords

• nonn
• tabf

Comments

Number of entries in row n is 1+floor(n/3).

Sum of entries in row n = A004148 (the RNA secondary structure numbers).

T(n,0)=A004149(n).

Sum(k*T(n,k),k>=0)=A110236(n-2) (n>=3).

Examples

			T(5,1)=4 because we have HHUHD, HUHDH, UHDH, and UUHDD.
Triangle starts:
1;
1;
1;
1,1;
2,2;
4,4;
8,8,1;
16,18,3;
		

Crossrefs

Cf. A004148, A004149, A110236

Programs

Maple

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

Formula

G.f. G=G(t,z) satisfies the equation G = 1 + zG + z^2*G(G-1-z+tz).