%I #17 Jan 03 2024 06:44:12
%S 1,1,1,1,1,2,2,1,1,1,2,2,4,5,5,5,3,1,1,1,2,2,4,5,8,10,12,13,15,17,16,
%T 13,9,4,1,1,1,2,2,4,5,8,10,15,18,23,27,34,40,47,52,56,57,57,56,50,39,
%U 26,14,5,1,1,1,2,2,4,5,8,10,15,18,26,32,42,50,63
%N Irregular triangle read by rows: number of {0,1,2}-shifted Schroeder paths of length n and area k.
%H Brian Drake, <a href="https://doi.org/10.1016/j.disc.2008.11.020">Limits of areas under lattice paths</a>, Discrete Math. 309 (2009), no. 12, 3936-3953. See Example 4.
%e Triangle begins
%e 1
%e 1 1
%e 1 1 2 2 1
%e 1 1 2 2 4 5 5 5 3 1
%e 1 1 2 2 4 5 8 10 12 13 15 17 16 13 9 4 1
%e ...
%t gf = Expand /@ FixedPoint[1 + x # + q x (1 + q x) # (Normal@# /. {x :> q^2 x}) + O[x]^7 &, 0];
%t Flatten[Reverse[CoefficientList[#, q]] & /@ CoefficientList[gf, x]] (* _Andrey Zabolotskiy_, Jan 03 2024 *)
%Y Row sums are A064641.
%Y Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201076 {0,2}, A201079 {0,2,4,6...}, A201080 {0,1,3,5...}.
%K nonn,tabf
%O 0,6
%A _N. J. A. Sloane_, Nov 27 2011
%E More terms from _Andrey Zabolotskiy_, Jan 03 2024