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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.

%I A201079 #18 Jan 03 2024 00:20:45
%S A201079 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,
%T A201079 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,
%U A201079 45,61,81,101,125,146,167,183,194,191,178,146,103,58,21,0,0,0
%N A201079 Irregular triangle read by rows: number of {0,2,4,6...}-shifted Schroeder paths of length n and area k.
%H A201079 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 A201079 Triangle begins
%e A201079 1
%e A201079 1
%e A201079 1 2 0
%e A201079 1 2 3 3 0
%e A201079 1 2 4 6 7 7 5 0 0
%e A201079 1 2 4 7 11 14 18 20 19 15 8 0 0
%e A201079 1 2 4 8 12 19 26 35 43 52 57 61 57 46 30 13 0 0 0
%e A201079 ...
%t A201079 max = 8; s0 = Range[2, max, 2];
%t A201079 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];
%t A201079 Flatten[Reverse[CoefficientList[#, q]][[;; ;; 2]] & /@ CoefficientList[gf, x]] (* _Andrey Zabolotskiy_, Jan 02 2024 *)
%Y A201079 Row sums give A063020. Rows converge to A015128.
%Y A201079 Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201076 {0,2}, A201080 {0,1,3,5...}, A201159 {0,1,2}.
%K A201079 nonn,tabf
%O A201079 0,4
%A A201079 _N. J. A. Sloane_, Nov 26 2011
%E A201079 Name and rows 3 and 5 corrected and row 7 added by _Andrey Zabolotskiy_, Jan 02 2024