A372878 a(n) is the sum of all symmetric valleys in the set of flattened Catalan words of length n.

1, 7, 33, 133, 496, 1770, 6142, 20902, 70107, 232489, 763927, 2491107, 8071234, 26007364, 83402988, 266351548, 847482277, 2687729595, 8499036925, 26804655025, 84336597636, 264777690382, 829636763338, 2594821366338, 8102197327711, 25259791668925, 78638974063827

Offset: 4

Author

Stefano Spezia, May 15 2024

Keywords

• nonn
• easy

Comments

The g.f. listed in Baril et al. has a mistake in the numerator: the factor (1 + 2*x) should be (1 - 2*x).

Links

• Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 18.
• Index entries for linear recurrences with constant coefficients, signature (9,-30,46,-33,9).

Crossrefs

Cf. A261064, A371963, A371964, A372875.

Programs

Mathematica

LinearRecurrence[{9,-30,46,-33,9},{1,7,33,133,496},28]

Formula

From Baril et al.: (Start)

G.f.: x^4*(1 - 2*x)/((1 - 3*x)^2*(1 - x)^3).

a(n) = (3^n*(2*n - 5) - 18*n^2 + 54*n - 27)/144. (End)

E.g.f.: (32 + exp(3*x)*(6*x - 5) - 9*exp(x)*(2*x^2 - 4*x + 3))/144.

a(n) - a(n-1) = A261064(n-3).