A182880 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,1)-steps. L_n is the set of lattice paths of weight n that start at (0,0) and 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, 1, 2, 3, 2, 5, 6, 8, 18, 13, 44, 6, 21, 102, 30, 34, 222, 120, 55, 466, 390, 20, 89, 948, 1140, 140, 144, 1884, 3066, 700, 233, 3672, 7770, 2800, 70, 377, 7044, 18780, 9800, 630, 610, 13332, 43710, 31080, 3780, 987, 24946, 98610, 91560, 17850, 252, 1597, 46218, 216732, 254400, 72450, 2772

Offset: 0

Author

Emeric Deutsch, Dec 11 2010

Keywords

• nonn
• tabf

Comments

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

Examples

			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, ud and du, have exactly one u step.
Triangle starts:
   1;
   1;
   2;
   3,  2;
   5,  6;
   8, 18;
  13, 44, 6;
		

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.

Crossrefs

Cf. A051286, A000045, A182881.

Programs

Maple

G:=1/sqrt(1-2*z-z^2+2*z^3+z^4-4*t*z^3): Gser:=simplify(series(G,z=0,18)): for n from 0 to 16 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 16 do seq(coeff(P[n],t,k),k=0..floor(n/3)) od; # yields sequence in triangular form

Formula

T(n,0) = A000045(n+1) (the Fibonacci numbers).

Sum_{k=0..n} k*T(n,k) = A182881(n).

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

The g.f. of column k is binomial(2n,n)*z^(3n)/(1-z-z^2)^(2n+1).

Apparently, T(n,1) = 2*A001628(n-3), T(n,2) = 6*A001873(n-6), T(n,3) = 20*A001875(n-9). - R. J. Mathar, Dec 11 2010