0,5

T(n,k) is the number of Łukasiewicz paths of length n having k steps (1,1). A Łukasiewicz path of length n is a path in the first quadrant from (0,0) to (n,0) using rise steps (1,k) for any positive integer k, level steps (1,0) and fall steps (1,-1) (see R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 6.19w; the integers are the slopes of the steps). Example: T(3,0)=2 because we have HHH and U(2)DD, where H=(1,0), U(2)=(1,2) and D=(1,-1). Row n has 1+floor(n/2) terms. Row sums yield the Catalan numbers (A000108). T(2n,n)=A000108(n). Column 0 is A102403

Alois P. Heinz, Rows n = 0..200, flattened

G.f.: G=G(t,z) satisfies z^3*(1-t)G^3+z(1-z+tz)G^2-G+1=0.

T(4,2) = 2 because we have UUDDUUDD and UUDUUDDD, where U=(1,1) and D=(1,-1).

Triangle begins:

1;

1;

1, 1;

2, 3;

6, 6, 2;

17, 15, 10;

m = 14; G[_, _] = 0;

Do[G[t_, z_] = 1 + G[t, z]^2 z + G[t, z]^2 t z^2 - G[t, z]^2 z^2 + G[t, z]^3 z^3 - G[t, z]^3 t z^3 + O[t]^m + O[z]^m, {m}];

CoefficientList[#, t]& /@ Take[CoefficientList[G[t, z], z], m] // Flatten (* Jean-François Alcover, Oct 05 2019 *)

Cf. A000108, A102403.

Sequence in context: A221020 A333304 A248896 * A246129 A359123 A124498

Adjacent sequences: A102399 A102400 A102401 * A102403 A102404 A102405

nonn,tabf

Emeric Deutsch, Jan 06 2005

approved