A372875 - OEIS

%I #16 May 17 2024 03:27:42

%S 1,2,5,13,1,34,7,90,31,1,242,113,10,659,375,59,1,1808,1189,271,13,

%T 4977,3686,1082,96,1,13715,11284,3976,534,16,37798,34239,13887,2507,

%U 142,1,104154,103115,46949,10555,929,19,286960,308452,155200,41324,5028,197,1

%N Irregular triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k symmetric valleys, with k >= 0.

%H Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, <a href="https://arxiv.org/abs/2405.05357">Flattened Catalan Words</a>, arXiv:2405.05357 [math.CO], 2024. See pp. 16-17.

%F G.f.: x*(1 - 2*x)*(1 - 2*x + 2*x^2 - x^2*y)/((1 - x)*(1 - 5*x + 8*x^2 - 5*x^3 - x^2*y + 2x^3*y)).

%F Sum_{k>=0} T(n,k) = A007051(n-1).

%e The irregular triangle begins:

%e      1;

%e      2;

%e      5;

%e     13,    1;

%e     34,    7;

%e     90,   31,    1;

%e    242,  113,   10;

%e    659,  375,   59,  1;

%e   1808, 1189,  271, 13;

%e   4977, 3686, 1082, 96, 1;

%e   ...

%e T(5,1) = 7 since there are 7 flattened Catalan words of length 5 with one symmetric valley: 00101, 01001, 01010, 01011, 01012, 01101, and 01212.

%t T[n_,k_]:=SeriesCoefficient[x(1-2x)(1-2x+2x^2-x^2y)/((1-x)(1-5x+8x^2-5x^3-x^2y+2x^3y)),{x,0,n},{y,0,k}];Join[{1},Flatten[ Table[T[n,k],{n,14},{k,0,Floor[(n-2)/2]}]]]

%Y Cf. A007051 (row sums), A371963, A371964, A372878.

%K nonn,tabf

%O 1,2

%A _Stefano Spezia_, May 15 2024