

A247470


Number of weak peaks in all weighted lattice paths in B(n).


1



0, 0, 0, 1, 4, 14, 43, 123, 337, 898, 2349, 6072, 15577, 39776, 101304, 257689, 655279, 1666772, 4242354, 10807191, 27557720, 70342486, 179736541, 459714008, 1176937542, 3015862454, 7734617111, 19852352861, 50992757233, 131071123062, 337122433947, 867624835207, 2234205069696
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OFFSET

0,5


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).
a(n) = Sum(k*A247469(n,k), k>=0).


LINKS

Table of n, a(n) for n=0..32.
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291306.


FORMULA

G.f. G = z^3*g/((1  z  z^2)^2*(1  z  z^2  2*z^3*g)), where g = 1 + z*g + z^2*g + z^3*g^2.


EXAMPLE

a(4) = 4 because B(4) = {hhhh, hhH, hHh, Hhh, HH, hu*d, u*h*d, u*dh} (weak peaks shown by *).


MAPLE

G := z^3*g/((1zz^2)^2*(1zz^22*z^3*g)): eq := g = 1+z*g+z^2*g+z^3*g^2: g := RootOf(eq, g): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);


CROSSREFS

Cf. A247469
Sequence in context: A135483 A095377 A296253 * A049539 A037528 A292718
Adjacent sequences: A247467 A247468 A247469 * A247471 A247472 A247473


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Sep 17 2014


STATUS

approved



