%I #8 May 17 2024 01:40:33

%S 1,2,6,20,67,222,728,2368,7653,24602,78730,250956,797159,2524342,

%T 7971612,25110584,78918985,247518642,774840974,2421378052,7554699531,

%U 23535794702,73222472416,227512682160,706073841197,2188828907722,6778308875538,20970393083708,64817578622383

%N a(n) is the total number of runs of weak ascents over all flattened Catalan words of length n.

%H 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 p. 10.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-22,24,-9).

%F From Baril et al.: (Start)

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

%F a(n) = (27 - 9*n + (5 + n)*3^n)/36. (End)

%F E.g.f.: (exp(3*x)*(5 + 3*x) - 9*exp(x)*(x - 3) - 32)/36.

%t LinearRecurrence[{8,-22,24,-9},{1,2,6,20},29]

%Y Cf. A372852, A372868.

%K nonn,easy

%O 1,2

%A _Stefano Spezia_, May 15 2024