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%I #5 Sep 17 2014 15:47:46
%S 1,1,2,4,8,17,36,80,180,410,946,2203,5173,12233,29108,69643,167437,
%T 404311,980125,2384441,5819576,14245384,34964611,86032272,212172668,
%U 524371704,1298509438,3221425567,8005623916,19926840746,49674610998,124006308008
%N Number of weighted lattice paths B(n) having no uudd strings.
%C 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.
%C a(n) = A247297(n,0).
%H M. Bona and A. Knopfmacher, <a href="http://dx.doi.org/10.1007/s00026-010-0060-7">On the probability that certain compositions have the same number of parts</a>, Ann. Comb., 14 (2010), 291-306.
%F G.f. G = G(z) satisfies G = 1 + z*G + z^2*G + z^3*G*(G - z^3 ).
%e a(6)=36 because among the 37 (=A004148(7)) paths in B(6) only uudd contains uudd.
%p 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);
%Y Cf. A004148, A247297.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Sep 17 2014
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