

A247288


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


2



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
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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/(1z) + t^2*z/(1t*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*(G1z/(1z)+t^2*z/(1t*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 .. n1) end do; # yields sequence in triangular form


CROSSREFS

Cf. A004148, A247289.
Sequence in context: A324642 A326757 A147787 * A135221 A318686 A214546
Adjacent sequences: A247285 A247286 A247287 * A247289 A247290 A247291


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Sep 14 2014


STATUS

approved



