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A201076 Irregular triangle read by rows: number of {0,2}-shifted Schroeder paths of length n and area k.

%I A201076 #15 Jan 03 2024 00:20:50
%S A201076 1,1,1,2,0,1,2,3,3,0,1,2,3,6,7,7,5,0,0,1,2,3,6,10,13,16,20,19,15,8,0,
%T A201076 0,1,2,3,6,10,16,22,29,39,48,53,56,57,46,30,13,0,0,0,1,2,3,6,10,16,25,
%U A201076 35,48,66,85,106,127,147,167,179,178,168,146,103,58,21,0,0,0
%N A201076 Irregular triangle read by rows: number of {0,2}-shifted Schroeder paths of length n and area k.
%H A201076 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.
%e A201076 Triangle begins
%e A201076 1
%e A201076 1
%e A201076 1 2 0
%e A201076 1 2 3 3 0
%e A201076 1 2 3 6 7 7 5 0 0
%e A201076 1 2 3 6 10 13 16 20 19 15 8 0 0
%e A201076 ...
%t A201076 gf = Expand /@ FixedPoint[1 + (q x + q^2 x^2) # (Normal@# /. {x :> q^2 x}) + O[x]^8 &, 0];
%t A201076 Flatten[Reverse[CoefficientList[#, q]][[;; ;; 2]] & /@ CoefficientList[gf, x]] (* _Andrey Zabolotskiy_, Jan 02 2024 *)
%Y A201076 Row sums give A052709. Rows converge to A101277.
%Y A201076 Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201079 {0,2,4,6...}, A201080 {0,1,3,5...}, A201159 {0,1,2}.
%K A201076 nonn,tabf
%O A201076 0,4
%A A201076 _N. J. A. Sloane_, Nov 26 2011
%E A201076 Row 5 corrected, rows 6-7 added by _Andrey Zabolotskiy_, Jan 02 2024