cp's OEIS Frontend

A372883 Irregular triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k symmetric peaks, with k >= 0.

%I A372883 #19 May 17 2024 19:49:10
%S A372883 1,2,4,1,9,5,23,17,1,63,51,8,176,149,39,1,491,439,153,11,1362,1308,
%T A372883 540,70,1,3762,3912,1812,342,14,10369,11671,5935,1439,110,1,28559,
%U A372883 34637,19175,5541,645,17,78653,102222,61302,20214,3170,159,1,216638,300190,194080,71242,13903,1089,20
%N A372883 Irregular triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k symmetric peaks, with k >= 0.
%H A372883 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. 21-22.
%F A372883 G.f.: x*(1 - x)*(1 - 2*x)/(1 - 5*x + 8*x^2 - 5*x^3 - x^2*y + 2*x^3*y).
%F A372883 Sum_{k>=0} T(n,k) = A007051(n-1).
%e A372883 The irregular triangle begins:
%e A372883      1;
%e A372883      2;
%e A372883      4,    1;
%e A372883      9,    5;
%e A372883     23,   17,    1;
%e A372883     63,   51,    8;
%e A372883    176,  149,   39,   1;
%e A372883    491,  439,  153,  11;
%e A372883   1362, 1308,  540,  70,  1;
%e A372883   3762, 3912, 1812, 342, 14;
%e A372883   ...
%e A372883 T(4,1) = 5 since there are 5 flattened Catalan words of length 4 with 1 symmetric peak: 0100, 0101, 0010, 0110, and 0121.
%t A372883 T[n_,k_]:=SeriesCoefficient[x(1-x)(1-2x)/(1-5x+8x^2-5x^3-x^2y+2x^3y),{x,0,n},{y,0,k}]; Table[T[n,k],{n,14},{k,0,Floor[(n-1)/2]}]//Flatten
%Y A372883 Cf. A007051 (row sums), A290900 (2nd column), A369328 (1st column), A371965, A372879, A372884.
%K A372883 nonn,tabf
%O A372883 1,2
%A A372883 _Stefano Spezia_, May 15 2024