A247298 Number of weighted lattice paths B(n) having no uudd strings.

1, 1, 2, 4, 8, 17, 36, 80, 180, 410, 946, 2203, 5173, 12233, 29108, 69643, 167437, 404311, 980125, 2384441, 5819576, 14245384, 34964611, 86032272, 212172668, 524371704, 1298509438, 3221425567, 8005623916, 19926840746, 49674610998, 124006308008

Offset: 0

Author

Emeric Deutsch, Sep 17 2014

Keywords

• nonn

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(n) = A247297(n,0).

Examples

			a(6)=36 because among the 37 (=A004148(7)) paths in B(6) only uudd contains uudd.
		

Links

• M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.

Crossrefs

Cf. A004148, A247297.

Programs

Maple

eq := G = 1+z*G+z^2*G+z^3*G*(G-z^3): G := RootOf(eq, G): Gser := series(G, z = 0, 37): seq(coeff(Gser, z, n), n = 0 .. 35);

Formula

G.f. G = G(z) satisfies G = 1 + z*G + z^2*G + z^3*G*(G - z^3 ).