A247288 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n having k weak peaks.

1, 1, 1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 2, 1, 1, 0, 8, 4, 3, 1, 1, 0, 16, 8, 7, 4, 1, 1, 0, 32, 16, 17, 10, 5, 1, 1, 0, 64, 32, 41, 26, 14, 6, 1, 1, 0, 128, 64, 98, 66, 39, 19, 7, 1, 1, 0, 256, 128, 232, 164, 107, 56, 25, 8, 1, 1, 0, 512, 256, 544, 400, 286, 164, 78, 32, 9, 1

Offset: 0

Author

Emeric Deutsch, Sep 14 2014

Keywords

• nonn
• tabf

Comments

A weak peak of a Motzkin path is a vertex on the top of a hump.

A hump is an upstep followed by 0 or more flatsteps followed by a downstep. For example, the peakless Motzkin path uhu*h*ddu*h*h*d where u=(1,1), h=(1,0), d(1,-1), has 5 weak peaks (shown by the stars).

Row n (n>=1) contains n entries.

Sum of entries in row n is the RNA secondary structure number A004148(n).

Sum(k*T(n,k), 0<=k<=n) = A247289(n).

Examples

			Row 4 is 1,0,2,1 because the peakless Motzkin paths hhhh, u*h*dhh, hu*h*dh, and u*h*h*d  have 0, 2, 2, and 3 weak peaks (shown by the stars).
Triangle starts:
1;
1;
1,0;
1,0,1;
1,0,2,1;
1,0,4,2,1;
1,0,8,4,3,1;
		

Links

• Alois P. Heinz, Rows n = 0..141, flattened

Crossrefs

Cf. A004148, A247289.

Programs

Maple

eq := G = 1+z*G+z^2*(G-1-z/(1-z)+t^2*z/(1-t*z))*G: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 14 do P[n] := sort(expand(coeff(Gser, z, n))) end do: 1; for n to 14 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form

Formula

The g.f. G(t,z) satisfies G = 1 + z*G + z^2*(G - 1 - z/(1-z) + t^2*z/(1-t*z))*G.