A201159 Irregular triangle read by rows: number of {0,1,2}-shifted Schroeder paths of length n and area k.

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 4, 5, 5, 5, 3, 1, 1, 1, 2, 2, 4, 5, 8, 10, 12, 13, 15, 17, 16, 13, 9, 4, 1, 1, 1, 2, 2, 4, 5, 8, 10, 15, 18, 23, 27, 34, 40, 47, 52, 56, 57, 57, 56, 50, 39, 26, 14, 5, 1, 1, 1, 2, 2, 4, 5, 8, 10, 15, 18, 26, 32, 42, 50, 63

Offset

0,6

Links

Table of n, a(n) for n=0..75.

Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953. See Example 4.

Example

Triangle begins
1
1 1
1 1 2 2 1
1 1 2 2 4 5 5 5 3 1
1 1 2 2 4 5 8 10 12 13 15 17 16 13 9 4 1
...

Mathematica

gf = Expand /@ FixedPoint[1 + x # + q x (1 + q x) # (Normal@# /. {x :> q^2 x}) + O[x]^7 &, 0];
Flatten[Reverse[CoefficientList[#, q]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 03 2024 *)

Crossrefs

Row sums are A064641.

Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201076 {0,2}, A201079 {0,2,4,6...}, A201080 {0,1,3,5...}.

Sequence in context: A280134 A351256 A143488 * A356225 A047070 A071127

Adjacent sequences: A201156 A201157 A201158 * A201160 A201161 A201162

Keyword

nonn,tabf

Author

N. J. A. Sloane, Nov 27 2011

Extensions

More terms from Andrey Zabolotskiy, Jan 03 2024

Status

approved