|
%I #16 Jan 03 2024 06:47:31
%S 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,
%T 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,
%U 5,1,1,1,1,2,3,4,5,7,10,13,16,21,26,31,37,45,54
%N Irregular triangle read by rows: number of Schroeder paths of length n and weighted area n^2-k.
%C 0 <= k <= n^2.
%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.
%e Triangle begins:
%e 1
%e 1 1
%e 1 1 1 2 1
%e 1 1 1 2 3 4 3 3 3 1
%e 1 1 1 2 3 4 5 7 8 9 10 11 10 7 6 4 1
%e ...
%t gf = Expand /@ FixedPoint[1 + x # (1 + q Normal@# /. {x :> q^2 x}) + O[x]^7 &, 0];
%t Flatten[Reverse[CoefficientList[#, q]] & /@ CoefficientList[gf, x]] (* _Andrey Zabolotskiy_, Jan 03 2024 *)
%Y Mirror image of A129179.
%Y Cf. A129176, A201076, A201079, A201080, A201159.
%K nonn,tabf
%O 0,7
%A _N. J. A. Sloane_, Nov 26 2011
%E More terms from _Andrey Zabolotskiy_, Jan 03 2024
|
