A372875 - OEIS

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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.

1, 2, 5, 13, 1, 34, 7, 90, 31, 1, 242, 113, 10, 659, 375, 59, 1, 1808, 1189, 271, 13, 4977, 3686, 1082, 96, 1, 13715, 11284, 3976, 534, 16, 37798, 34239, 13887, 2507, 142, 1, 104154, 103115, 46949, 10555, 929, 19, 286960, 308452, 155200, 41324, 5028, 197, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..50.

Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See pp. 16-17.

FORMULA

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)).

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

EXAMPLE

The irregular triangle begins:

13, 1;

34, 7;

90, 31, 1;

242, 113, 10;

659, 375, 59, 1;

1808, 1189, 271, 13;

4977, 3686, 1082, 96, 1;

...

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.

MATHEMATICA

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]}]]]

CROSSREFS

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

Sequence in context: A042241 A042911 A137918 * A114502 A135308 A114492

Adjacent sequences: A372872 A372873 A372874 * A372876 A372877 A372878

KEYWORD

nonn,tabf

AUTHOR

Stefano Spezia, May 15 2024

STATUS

approved