A201079 Irregular triangle read by rows: number of {0,2,4,6...}-shifted Schroeder paths of length n and area k.

1, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 2, 4, 6, 7, 7, 5, 0, 0, 1, 2, 4, 7, 11, 14, 18, 20, 19, 15, 8, 0, 0, 1, 2, 4, 8, 12, 19, 26, 35, 43, 52, 57, 61, 57, 46, 30, 13, 0, 0, 0, 1, 2, 4, 8, 13, 21, 32, 45, 61, 81, 101, 125, 146, 167, 183, 194, 191, 178, 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 4 6 7 7 5 0 0
1 2 4 7 11 14 18 20 19 15 8 0 0
1 2 4 8 12 19 26 35 43 52 57 61 57 46 30 13 0 0 0
...
		

Links

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

Crossrefs

Row sums give A063020. Rows converge to A015128.

Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201076 {0,2}, A201080 {0,1,3,5...}, A201159 {0,1,2}.

Programs

Mathematica

max = 8; s0 = Range[2, max, 2];
gf = Expand /@ FixedPoint[With[{g = Normal@#}, 1 + q x g (g /. {x :> q^2 x}) + Sum[q^(j^2 - j) x^j Product[g /. {x :> q^(2 i - 2) x}, {i, j}], {j, s0}] + O[x]^max] &, 0];
Flatten[Reverse[CoefficientList[#, q]][[;; ;; 2]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 02 2024 *)

Extensions

Name and rows 3 and 5 corrected and row 7 added by Andrey Zabolotskiy, Jan 02 2024