OFFSET
0,3
COMMENTS
B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps.
A weak peak in a lattice 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 weighted lattice path Hu*duu*h*H*dd has 4 weak peaks (shown by the stars).
Row n contains n-1 entries (n>=2).
Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
T(n,0) = A000045(n+1) (the Fibonacci numbers).
Sum(k*T(n,k), k=0..n) = A247470(n).
LINKS
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
FORMULA
G.f. G = G(t,z) satisfies G = 1 + z*G + z^2*G + z^3*G*(G - 1/(1 - z - z^2) + t/(1 - t*z - t*z^2)).
EXAMPLE
Row 3 is 3, 1 because B(3) = {u*d, hH, Hh, hhh} (weak peaks shown by *).
Triangle starts:
1;
1;
2;
3,1;
5,2,1;
8,5,3,1;
13,11,8,4,1;
MAPLE
eq := G = 1+z*G+z^2*G+z^3*G*(G-1/(1-z-z^2)+t/(1-t*z-t*z^2)): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 17 do seq(coeff(P[n], t, k), k = 0 .. n-2) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 17 2014
STATUS
approved