A182886 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.

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, 60, 34, 10, 6, 1, 0, 30, 90, 60, 121, 52, 15, 7, 1, 20, 0, 180, 215, 76, 21, 8, 1, 0, 140, 90, 630, 540, 421, 351, 107, 28, 9, 1, 0, 140, 560, 630, 1710, 1176, 846, 539, 146, 36, 10, 1

Offset

0,7

Comments

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

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

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

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

Links

Table of n, a(n) for n=0..75.

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

E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Formula

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).

Example

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.
Triangle starts:
  1;
  0,  1;
  0,  1,  1;
  2,  0,  2,  1;
  0,  6,  1,  3,  1;
  0,  6, 12,  3,  4,  1;

Maple

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

Crossrefs

Cf. A051286, A182887, A000984.

Sequence in context: A362614 A336167 A076626 * A108731 A235168 A060950

Adjacent sequences: A182883 A182884 A182885 * A182887 A182888 A182889

Keyword

nonn,tabl

Author

Emeric Deutsch, Dec 11 2010

Status

approved