|
%I #29 Jan 17 2024 10:35:11
%S 1,1,8,49,246,1157,5248,23256,101398,436865,1865136,7906054,33319388,
%T 139754994,583859968,2430991670,10092510630,41794856985,172699266480,
%U 712220712390,2932169392020,12052941519030,49475929052160,202838118604680
%N a(n) = 2^(2*n - 5)*binomial(n-5/2, -1/2)*(36*n^4 - 78*n^3 + 54*n^2 - 48*n + 24)/((n + 1)*n*(n - 1)) for n >= 2 and otherwise 1.
%C Conjecture: These are the number of linear intervals in the Cambrian lattices of type D_n. An interval is linear if it is isomorphic to a total order. The conjecture has been checked up to the term a(8) = 101398.
%C The term a(3) = 49 is the same as the 49 appearing in A344136.
%H 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. 152.
%H Peter Luschny, <a href="/A344400/a344400.pdf">Remark regarding A344228 and A344321</a>.
%F a(n) = (3*n-2)*(1/n+1/2)*binomial(2*n-2,n-1) + 6*(n-2)*binomial(2*n-4,n-2) + (n-1)*(3*n-8)/(2*(2*n-3))*binomial(2*n-2,n-1) + 2 Sum_{k=1..2n-6} binomial(k,n-1)*(n+1+k) for n >= 3.
%F a(n) = A344401(n) / A007531(n+3) for n >= 2. - _Peter Luschny_, May 17 2021
%p a := n -> if n < 2 then 1 else 2^(2*n - 5)*binomial(n - 5/2, -1/2)*(36*n^4 - 78*n^3 + 54*n^2 - 48*n + 24)/((n + 1)*n*(n - 1)) fi;
%p seq(a(n), n = 0..23); # _Peter Luschny_, May 16 2021
%o (Sage)
%o def a(n):
%o if n < 2: return 1
%o if n == 2: return 8
%o return (3*n-2)*(1/n+1/2)*binomial(2*n-2,n-1)+6*(n-2)*binomial(2*n-4,n-2)+(n-1)*(3*n-8)/2/(2*n-3)*binomial(2*n-2,n-1)+sum(2*binomial(k,n-1)*(n+1+k) for k in range(n-1,2*n-5))
%o print([a(n) for n in range(24)])
%Y Cf. A344136 for the type A, A344228 for the type B.
%Y Cf. also A344191, A344216 for similar sequences.
%Y Cf. A344400 and A344401 for an alternative approach.
%Y Cf. A007531.
%K nonn
%O 0,3
%A _F. Chapoton_, May 15 2021
%E Better name from _Peter Luschny_, May 16 2021
| 