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

0,10

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).

Links

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

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.

Example

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;

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

Crossrefs

Cf. A004148, A247289.

Sequence in context: A371740 A326757 A147787 * A135221 A318686 A214546

Adjacent sequences: A247285 A247286 A247287 * A247289 A247290 A247291

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 14 2014

Status

approved