A102404 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n having k ascents of length 1 starting at an even level.

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 5, 5, 3, 0, 1, 14, 14, 9, 4, 0, 1, 39, 46, 27, 14, 5, 0, 1, 114, 143, 101, 44, 20, 6, 0, 1, 339, 466, 341, 184, 65, 27, 7, 0, 1, 1028, 1524, 1212, 664, 300, 90, 35, 8, 0, 1, 3163, 5043, 4279, 2539, 1145, 454, 119, 44, 9, 0, 1, 9852, 16812, 15206, 9564, 4665, 1819, 651, 152, 54, 10, 0, 1

Offset: 0

Author

Emeric Deutsch, Jan 06 2005

Keywords

• nonn
• tabl

Comments

T(n,k) is number of Łukasiewicz paths of length n having k level steps at an even level. 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(4,1)=5 because we have (H)UHD, (H)U(2)DD, UHD(H), U(2)DD(H) and U(2)(H)DD, where H=(1,0), U(2)=(1,2) and D=(1,-1) and the level steps at even level are shown between parentheses. Row n contains n+1 terms. Row sums yield the Catalan numbers (A000108). Column 0 is A102406.

Examples

			T(3,1)=2 because we have (U)DUUDD and UUDD(U)D, where U=(1,1), D=(1,-1) and the ascents of length 1 starting at an even level are shown between parentheses.
		

Links

• Index entries for sequences related to Łukasiewicz

Crossrefs

Cf. A000108, A102405, A102406.

Programs

Mathematica

m = 12; G[_] = 0;
Do[G[z_] = -((G[z]^2 z ((t-1)z - 1)^2 + 1)/((t-1)z^2 + (t-1)z - 1)) + O[z]^m, {m}];
CoefficientList[#, t]& /@ CoefficientList[G[z], z] // Flatten (* Jean-François Alcover, Nov 15 2019 *)

Formula

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