cp's OEIS Frontend

A372875 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.

%I A372875 #16 May 17 2024 03:27:42
%S A372875 1,2,5,13,1,34,7,90,31,1,242,113,10,659,375,59,1,1808,1189,271,13,
%T A372875 4977,3686,1082,96,1,13715,11284,3976,534,16,37798,34239,13887,2507,
%U A372875 142,1,104154,103115,46949,10555,929,19,286960,308452,155200,41324,5028,197,1
%N A372875 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 A372875 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 A372875 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 A372875 Sum_{k>=0} T(n,k) = A007051(n-1).
%e A372875 The irregular triangle begins:
%e A372875      1;
%e A372875      2;
%e A372875      5;
%e A372875     13,    1;
%e A372875     34,    7;
%e A372875     90,   31,    1;
%e A372875    242,  113,   10;
%e A372875    659,  375,   59,  1;
%e A372875   1808, 1189,  271, 13;
%e A372875   4977, 3686, 1082, 96, 1;
%e A372875   ...
%e A372875 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 A372875 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 A372875 Cf. A007051 (row sums), A371963, A371964, A372878.
%K A372875 nonn,tabf
%O A372875 1,2
%A A372875 _Stefano Spezia_, May 15 2024