A182882 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps of weight 1. L_n is the set of lattice paths of weight n that start at (0,0) , end on 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; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 1, 6, 3, 0, 1, 6, 3, 12, 4, 0, 1, 7, 24, 6, 20, 5, 0, 1, 12, 34, 60, 10, 30, 6, 0, 1, 31, 60, 100, 120, 15, 42, 7, 0, 1, 40, 185, 180, 230, 210, 21, 56, 8, 0, 1, 91, 260, 645, 420, 455, 336, 28, 72, 9, 0, 1, 170, 636, 980, 1715, 840, 812, 504, 36, 90, 10, 0, 1

Offset

0,7

Comments

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

T(n,0) = A182883(n).

Sum_{k=0..n} k*T(n,k) = A182884(n).

References

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.

Links

Table of n, a(n) for n=0..77.

Formula

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

Example

T(3,1)=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 have exactly one h step.
Triangle starts:
  1;
  0, 1;
  1, 0,  1;
  2, 2,  0, 1;
  1, 6,  3, 0, 1;
  6, 3, 12, 4, 0, 1;
  ...

Maple

G:=1/sqrt(1-2*t*z-2*z^2+t^2*z^2+2*t*z^3+z^4-4*z^3): Gser:=simplify(series(G, z=0, 15)): 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, A182883, A182884.

Sequence in context: A104579 A390644 A079531 * A134178 A059018 A249371

Adjacent sequences: A182879 A182880 A182881 * A182883 A182884 A182885

Keyword

nonn,tabl

Author

Emeric Deutsch, Dec 11 2010

Status

approved