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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
0 <= k <= n^2.
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 *)
nonn,tabf,more,changed
More terms from Andrey Zabolotskiy, Jan 03 2024
Brian Drake, <a href="httphttps://dx.doi.org/10.1016/j.disc.2008.11.020">Limits of areas under lattice paths</a>, Discrete Math. 309 (2009), no. 12, 3936-3953.
Mirror image of A129179.
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