A102402 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2.

1, 1, 1, 1, 2, 3, 6, 6, 2, 17, 15, 10, 46, 51, 30, 5, 128, 175, 91, 35, 372, 568, 336, 140, 14, 1109, 1827, 1296, 504, 126, 3349, 5980, 4785, 2010, 630, 42, 10221, 19833, 17215, 8415, 2640, 462, 31527, 66078, 61908, 34210, 11385, 2772, 132, 98178, 220649, 223444, 134706, 50908, 13299, 1716

Offset: 0

Author

Emeric Deutsch, Jan 06 2005

Keywords

• nonn
• tabf

Comments

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

Examples

			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;
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• Index entries for sequences related to Łukasiewicz

Crossrefs

Cf. A000108, A102403.

Programs

Mathematica

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 *)

Formula

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