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

0,4

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 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
...

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 *)

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}.

Sequence in context: A214568 A107017 A201076 * A318601 A306814 A241382

Adjacent sequences: A201076 A201077 A201078 * A201080 A201081 A201082

Keyword

nonn,tabf

Author

N. J. A. Sloane, Nov 26 2011

Extensions

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

Status

approved