A182886 - OEIS

%I #10 Nov 04 2019 02:24:57

%S 1,0,1,0,1,1,2,0,2,1,0,6,1,3,1,0,6,12,3,4,1,6,0,24,21,6,5,1,0,30,12,

%T 60,34,10,6,1,0,30,90,60,121,52,15,7,1,20,0,180,215,76,21,8,1,0,140,

%U 90,630,540,421,351,107,28,9,1,0,140,560,630,1710,1176,846,539,146,36,10,1

%N Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps. These are paths that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1; a (1,0)-step with weight 2; a (1,1)-step with weight 2; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

%C Sum of entries in row n is A051286(n).

%C T(3n,0) = binomial(2n,n) = A000984(n).

%C T(n,0) = 0 if n mod 3 > 0.

%C Sum_{k=0..n} k*T(n,k) = A182887(n).

%H M. Bona and A. Knopfmacher, <a href="https://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.

%H E. Munarini, N. Zagaglia Salvi, <a href="https://doi.org/10.1016/S0012-365X(02)00378-3">On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns</a>, Discrete Mathematics 259 (2002), 163-177.

%F G.f.: G(t,z) = 1/sqrt(1 - 2tz - 2tz^2 + t^2*z^2 + 2t^2*z^3 + t^2*z^4 - 4z^3).

%e T(3,2)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them, namely hH and Hh, have exactly two (1,0)-steps.

%e Triangle starts:

%e   1;

%e   0,  1;

%e   0,  1,  1;

%e   2,  0,  2,  1;

%e   0,  6,  1,  3,  1;

%e   0,  6, 12,  3,  4,  1;

%p G:=1/sqrt(1-2*t*z-2*t*z^2+t^2*z^2+2*t^2*z^3+t^2*z^4-4*z^3): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,k),k=0..n) od; # yields sequence in triangular form

%Y Cf. A051286, A182887, A000984.

%K nonn,tabl

%O 0,7

%A _Emeric Deutsch_, Dec 11 2010