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A201079 Irregular triangle read by rows: number of {0,2,4,6...}-shifted Schroeder paths of length n and area k. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
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
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

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)