A166293 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n and having k peaks at even level (0<=k<=n-1; U=(1,1), D=(1,-1)).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 12, 5, 1, 1, 6, 22, 28, 18, 6, 1, 1, 7, 35, 59, 50, 25, 7, 1, 1, 8, 54, 114, 124, 80, 33, 8, 1, 1, 9, 82, 210, 279, 226, 119, 42, 9, 1, 1, 10, 124, 374, 592, 576, 375, 168, 52, 10, 1, 1, 11, 188, 653, 1199, 1374, 1062, 582, 228, 63

Offset: 1

Author

Emeric Deutsch, Oct 12 2009

Keywords

• nonn
• tabl

Comments

Sum of entries in row n = A004148(n+1) (the secondary structure numbers).

T(n,0)=1.

Sum(k*T(n,k), k=0..n-1)=A166294(n).

Examples

			T(4,2)=3 because we have UDU(UD)(UD)D, U(UD)(UD)DUD, and U(UD)DU(UD)D (the even-level peaks are shown between parentheses).
Triangle starts:
1;
1,1;
1,2,1;
1,3,3,1;
1,4,7,4,1;
1,5,13,12,5,1.
		

Crossrefs

Cf. A004148, A166291, A166292, A166294

Programs

Maple

p1 := -G+1+t*z*G+s*z^2*G+s^2*z^3*H*G: p2 := subs({t = s, s = t, G = H, H = G}, p1): r := resultant(p1, p2, H): G := RootOf(subs(t = 1, r), G): Gser := simplify(series(G, z = 0, 15)): for n to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 12 do seq(coeff(P[n], s, j), j = 0 .. n-1) end do; # yields sequence in triangular form

Formula

The trivariate g.f. G=G(t,s,z), where z marks semilength, t marks odd-level peaks and s marks even-level peaks, satisfies G = 1 + tzG + sz^2*G + s^2*z^3*HG, where H=G(s,t,z) (interchanging t and s and eliminating H, one obtains G(t,s,z); see the Maple program).