A372884 -id:A372884 - 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.

1, 2, 4, 1, 9, 5, 23, 17, 1, 63, 51, 8, 176, 149, 39, 1, 491, 439, 153, 11, 1362, 1308, 540, 70, 1, 3762, 3912, 1812, 342, 14, 10369, 11671, 5935, 1439, 110, 1, 28559, 34637, 19175, 5541, 645, 17, 78653, 102222, 61302, 20214, 3170, 159, 1, 216638, 300190, 194080, 71242, 13903, 1089, 20

Offset: 1

Stefano Spezia, May 15 2024

			The irregular triangle begins:
     1;
     2;
     4,    1;
     9,    5;
    23,   17,    1;
    63,   51,    8;
   176,  149,   39,   1;
   491,  439,  153,  11;
  1362, 1308,  540,  70,  1;
  3762, 3912, 1812, 342, 14;
  ...
T(4,1) = 5 since there are 5 flattened Catalan words of length 4 with 1 symmetric peak: 0100, 0101, 0010, 0110, and 0121.

Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See pp. 21-22.

Cf. A007051 (row sums), A290900 (2nd column), A369328 (1st column), A371965, A372879, A372884.

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

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

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

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.

Original entry on oeis.org

1, 2, 4, 1, 9, 5, 22, 18, 1, 56, 58, 8, 145, 178, 41, 1, 378, 532, 173, 11, 988, 1563, 656, 73, 1, 2585, 4535, 2327, 381, 14, 6766, 13030, 7888, 1726, 114, 1, 17712, 37140, 25872, 7124, 709, 17, 46369, 105156, 82758, 27534, 3739, 164, 1, 121394, 296040, 259542, 101350, 17632, 1184, 20

Offset: 1

Stefano Spezia, May 15 2024

			The irregular triangle begins:
    1;
    2;
    4,    1;
    9,    5;
   22,   18,   1;
   56,   58,   8;
  145,  178,  41,  1;
  378,  532, 173, 11;
  988, 1563, 656, 73, 1;
  ...
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.

Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See pp. 19-20.

Cf. A007051 (row sums), A055588, A371965, A372883, A372884.

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

G.f.: x*(1 - 2*x)/((1 - x)*(1 - 3*x + x^2*(1 - y))).
T(n,0) = A055588(n-1).
Sum_{k>=0} T(n,k) = A007051(n-1).

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.

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