A201075 Irregular triangle read by rows: number of Schroeder paths of length n and weighted area n^2-k.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 4, 3, 3, 3, 1, 1, 1, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 10, 7, 6, 4, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 14, 17, 22, 25, 27, 31, 34, 34, 33, 31, 28, 21, 14, 10, 5, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 26, 31, 37, 45, 54

Offset: 0

Author

N. J. A. Sloane, Nov 26 2011

Keywords

• nonn
• tabf

Comments

0 <= k <= n^2.

Examples

			Triangle begins:
1
1 1
1 1 1 2 1
1 1 1 2 3 4 3 3 3 1
1 1 1 2 3 4 5 7 8 9 10 11 10 7 6 4 1
...
		

Links

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

Crossrefs

Mirror image of A129179.

Cf. A129176, A201076, A201079, A201080, A201159.

Programs

Mathematica

gf = Expand /@ FixedPoint[1 + x # (1 + q Normal@# /. {x :> q^2 x}) + O[x]^7 &, 0];
Flatten[Reverse[CoefficientList[#, q]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 03 2024 *)

Extensions

More terms from Andrey Zabolotskiy, Jan 03 2024