A201076 Irregular triangle read by rows: number of {0,2}-shifted Schroeder paths of length n and area k.

1, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 2, 3, 6, 7, 7, 5, 0, 0, 1, 2, 3, 6, 10, 13, 16, 20, 19, 15, 8, 0, 0, 1, 2, 3, 6, 10, 16, 22, 29, 39, 48, 53, 56, 57, 46, 30, 13, 0, 0, 0, 1, 2, 3, 6, 10, 16, 25, 35, 48, 66, 85, 106, 127, 147, 167, 179, 178, 168, 146, 103, 58, 21, 0, 0, 0

Offset

0,4

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.

Example

Triangle begins
1
1
1 2 0
1 2 3 3 0
1 2 3 6 7 7 5 0 0
1 2 3 6 10 13 16 20 19 15 8 0 0
...

Mathematica

gf = Expand /@ FixedPoint[1 + (q x + q^2 x^2) # (Normal@# /. {x :> q^2 x}) + O[x]^8 &, 0];
Flatten[Reverse[CoefficientList[#, q]][[;; ;; 2]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 02 2024 *)

Crossrefs

Row sums give A052709. Rows converge to A101277.

Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201079 {0,2,4,6...}, A201080 {0,1,3,5...}, A201159 {0,1,2}.

Sequence in context: A119464 A214568 A107017 * A201079 A318601 A306814

Adjacent sequences: A201073 A201074 A201075 * A201077 A201078 A201079

Keyword

nonn,tabf

Author

N. J. A. Sloane, Nov 26 2011

Extensions

Row 5 corrected, rows 6-7 added by Andrey Zabolotskiy, Jan 02 2024

Status

approved