%I #5 Mar 30 2012 17:36:25
%N Triangle read by rows: T(n,k) is the number of weighted lattice paths in B(n) having k valleys. The members of B(n) are paths of weight n that start at (0,0), end on but never go below the horizontal axis, and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps. A valley is a (1,-1)-step followed by a (1,1)-step.
%C Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
%C Sum(k*T(n,k), k>=0) = A182902(n).
%C For the distribution of the statistic "number of peaks" see A162984. A peak is a (1,1)-step followed by a (1,-1)-step.
%D M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
%F G. f.: F=F(v,z) satisfies z^3*(z+z^2+v-vz-vz^2)F^2 - (1-z-z^2-z^3+vz^3)F+1=0 (z marks weight, v marks number of valleys).
%F The trivariate g.f. H(u,v,z), where u (v) marks peaks (valleys) and z marks weight is given by H=1+zH+z^2*H+z^3*(u-1+H)[v(H-1-zH-z^2*H)+1+zH+z^2*H].
%e T(7,1)=4. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), we have hUDUD, UDUDh, UDUhD, and UhDUD.
%e Triangle starts:
%p eq := z^3*(z+z^2+v-v*z-v*z^2)*F^2-(1-z-z^2-z^3+v*z^3)*F+1 = 0: F := RootOf(eq, F): Fser := simplify(series(F, z = 0, 20)): for n from 0 to 18 do P[n] := sort(coeff(Fser, z, n)) end do: 1; 1; 2; for n from 0 to 18 do seq(coeff(P[n], v, k), k = 0 .. floor((1/3)*n)-1) end do; # yields sequence in triangular form
%Y Cf. A004148, A182901, A182902, A162984.
%A _Emeric Deutsch_, Dec 15 2010