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A372879 Irregular triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k short peak, with k >= 0.

%I A372879 #18 May 17 2024 03:27:57
%S A372879 1,2,4,1,9,5,22,18,1,56,58,8,145,178,41,1,378,532,173,11,988,1563,656,
%T A372879 73,1,2585,4535,2327,381,14,6766,13030,7888,1726,114,1,17712,37140,
%U A372879 25872,7124,709,17,46369,105156,82758,27534,3739,164,1,121394,296040,259542,101350,17632,1184,20
%N A372879 Irregular triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k short peak, with k >= 0.
%H A372879 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. 19-20.
%F A372879 G.f.: x*(1 - 2*x)/((1 - x)*(1 - 3*x + x^2*(1 - y))).
%F A372879 T(n,0) = A055588(n-1).
%F A372879 Sum_{k>=0} T(n,k) = A007051(n-1).
%e A372879 The irregular triangle begins:
%e A372879     1;
%e A372879     2;
%e A372879     4,    1;
%e A372879     9,    5;
%e A372879    22,   18,   1;
%e A372879    56,   58,   8;
%e A372879   145,  178,  41,  1;
%e A372879   378,  532, 173, 11;
%e A372879   988, 1563, 656, 73, 1;
%e A372879   ...
%e A372879 T(6,2) = 8 since there are 8 flattened Catalan words of length 6 with 2 short peaks: 001010, 010100, 010101, 010010, 010120, 010121, 012010, and 012121.
%t A372879 T[n_,k_]:=SeriesCoefficient[x(1-2x)/((1-x)(1-3x+x^2(1-y))),{x,0,n},{y,0,k}]; Table[T[n,k],{n,14},{k,0,Floor[(n-1)/2]}]//Flatten
%Y A372879 Cf. A007051 (row sums), A055588, A371965, A372883, A372884.
%K A372879 nonn,tabf
%O A372879 1,2
%A A372879 _Stefano Spezia_, May 15 2024