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

0,5

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

Links

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

Index entries for sequences related to Łukasiewicz

Formula

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

Example

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;

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 *)

Crossrefs

Cf. A000108, A102403.

Sequence in context: A221020 A333304 A248896 * A246129 A359123 A124498

Adjacent sequences: A102399 A102400 A102401 * A102403 A102404 A102405

Keyword

nonn,tabf

Author

Emeric Deutsch, Jan 06 2005

Status

approved