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A344228 a(n) = binomial(2*n,n)*(2*n+1)/2+n*binomial(2*n-2,n)+(n-1)*binomial(2*n-2,n+1).

%I A344228 #23 Jan 17 2024 10:34:55
%S A344228 3,17,84,393,1778,7866,34254,147433,628914,2663934,11219728,47033322,
%T A344228 196393044,817338580,3391858530,14040986985,57998364690,239112756630,
%U A344228 984126777480,4044255577230,16597080112860,68027923573740
%N A344228 a(n) = binomial(2*n,n)*(2*n+1)/2+n*binomial(2*n-2,n)+(n-1)*binomial(2*n-2,n+1).
%C A344228 Conjecture: These are the number of linear intervals in the Cambrian lattices of type B_n. An interval is linear if it is isomorphic to a total order. The conjecture has been checked up to the term 34254 for n = 7.
%H A344228 Michael De Vlieger, <a href="/A344228/b344228.txt">Table of n, a(n) for n = 1..1658</a>
%H A344228 Clément Chenevière, <a href="https://theses.hal.science/tel-04255439">Enumerative study of intervals in lattices of Tamari type</a>, Ph. D. thesis, Univ. Strasbourg (France), Ruhr-Univ. Bochum (Germany), HAL tel-04255439 [math.CO], 2024. See p. 151.
%F A344228 From _Peter Luschny_, May 12 2021: (Start)
%F A344228 a(n) = 3*(2*n^3 + n - 1)*2^(2*n - 2)*binomial(n - 3/2, -1/2)/((n + 1)*n).
%F A344228 a(n) = [x^n] (15*x - 24*x^2 + 8*x^3 - 2 + (1 - 4*x)^(3/2)*(2 - 3*x))/(2*(1 - 4*x)^(3/2)*x).
%F A344228 a(n) ~ 4^(n-2)*(24*n - 15)/sqrt(Pi*n). (End)
%F A344228 a(n) = a(n-1)*2*(2*n - 3)*(2*n^3 + n - 1)/((n + 1)*(2*n^3 - 6*n^2 + 7*n - 4)) for n > 1. - _Chai Wah Wu_, May 13 2021
%e A344228 For B_2, among the 18 intervals in the hexagon-shaped lattice, only one is not linear.
%p A344228 a := n -> 3*(2*n^3 + n - 1)*2^(2*n - 2)*binomial(n - 3/2, -1/2)/((n + 1)*n):
%p A344228 seq(a(n), n = 1..22);  # _Peter Luschny_, May 12 2021
%t A344228 Array[3 (2 #^3 + # - 1)*2^(2 # - 2)*Binomial[# - 3/2, -1/2]/(# (# + 1)) &, 22] (* _Michael De Vlieger_, Jan 17 2024 *)
%o A344228 (Sage)
%o A344228 def a(n):
%o A344228     return binomial(2*n,n)*(2*n+1)/2+n*binomial(2*n-2,n)+(n-1)*binomial(2*n-2,n+1)
%Y A344228 Cf. A344136 for the type A, A344191 for a similar sequence.
%K A344228 nonn
%O A344228 1,1
%A A344228 _F. Chapoton_, May 12 2021