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

Author

N. J. A. Sloane, Nov 26 2011

Keywords

• nonn
• tabf

Examples

			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
...
		

Links

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

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}.

Programs

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 *)

Extensions

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