A372868 - OEIS

A372868

Irregular triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of weak ascents, with 1 <= k <= ceiling(n/2).

1, 2, 4, 1, 8, 6, 16, 24, 1, 32, 80, 10, 64, 240, 60, 1, 128, 672, 280, 14, 256, 1792, 1120, 112, 1, 512, 4608, 4032, 672, 18, 1024, 11520, 13440, 3360, 180, 1, 2048, 28160, 42240, 14784, 1320, 22, 4096, 67584, 126720, 59136, 7920, 264, 1, 8192, 159744, 366080, 219648, 41184, 2288, 26

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OFFSET

1,2

COMMENTS

With offset 0 for the variable k, T(n,k) is the number of flattened Catalan words of length n with exactly k peaks. In such case, T(4,1) = 6 corresponds to 6 flattened Catalan words of length 4 with 1 peak: 0010, 0100, 0110, 0101, 0120, and 0121. See Baril et al. at page 20.

LINKS

Table of n, a(n) for n=1..56.

Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See pp. 8-9.

FORMULA

G.f.: (1-2*x)*x*y/(1 - 4*x + 4*x^2 - x^2*y).

T(n,k) = 2^(n-2*k+1)*binomial(n-1, 2*k-2).

T(n,1) = A000079(n-1).

T(n,2) = A001788(n-2).

T(n,3) = A003472(n-1).

T(n,4) = A002409(n-7).

T(n,5) = A140325(n-9).

T(n,6) = A172242(n-1).