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
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 14 2014
STATUS
approved