%I #76 Oct 05 2020 23:34:18

%S 0,1,2,8,41,240,1528,10312,72647,528992,3954488,30201504,234798627,

%T 1853076528,14814453896,119763949936,977709717091,8050816106176,

%U 66803956281592,558146870481760,4692269111973668,39669049950811328,337082395954643168,2877697636252004168,24672447821197834553

%N Number of indecomposable intervals in the Tamari lattices.

%C This is also the number of interval-posets with connected Hasse diagram.

%H Alois P. Heinz, <a href="/A294084/b294084.txt">Table of n, a(n) for n = 0..1031</a>

%H F. Chapoton, <a href="https://arxiv.org/abs/math/0602368">Sur le nombre d'intervalles dans les treillis de Tamari</a>, arXiv:math/0602368 [math.CO], 2006.

%H F. Chapoton, <a href="http://www.kurims.kyoto-u.ac.jp/EMIS/journals/SLC/wpapers/s55chapoton.html">Sur le nombre d'intervalles dans les treillis de Tamari</a>, Séminaire Lotharingien de combinatoire, vol. 55 (2006).

%H Alois Panholzer, <a href="https://arxiv.org/abs/2007.14676">Parking function varieties for combinatorial tree models</a>, arXiv:2007.14676 [math.CO], 2020.

%F The generating series can be obtained by inverting the generating series of A000260.

%e Among the 3 interval-posets of size 2 :

%e 1 --> 2 ; 1 <-- 2 ; 1 2,

%e only the third (which is an antichain) is not a connected poset.

%p h:= proc(n) h(n):= 2*(4*n+1)!/((n+1)!*(3*n+2)!) end:

%p a:= proc(n) a(n):= `if`(n=0, 0, h(n)-add(a(n-i)*h(i), i=1..n-1)) end:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Feb 28 2019

%t terms = 25;

%t G[_] = 0; Do[G[x_] = 1 + x G[x]^4 + O[x]^terms, terms];

%t F[x_] = 1 - 1/((2 - G[x]) G[x]^2);

%t CoefficientList[F[x], x] (* _Jean-François Alcover_, Feb 14 2019 *)

%o (Sage)

%o F = PowerSeriesRing(ZZ,'t')([1] + [(2 * binomial(4 * n + 1, n - 1)) // (n * (n + 1)) for n in range(1, 20)])

%o 1 - F.inverse()

%o (Julia)

%o using Nemo

%o s(n) = div(Nemo.binom(4*n + 2, n + 1), (2*n + 1) * (3*n + 2))

%o R, z = PowerSeriesRing(ZZ, 25, "z")

%o F = sum(s(n)z^n for n in 0:25)

%o G = 1 - inv(F)

%o println([coeff(G,n) for n in 0:24]) # _Peter Luschny_, Feb 26 2018

%Y Cf. A000260.

%K nonn

%O 0,3

%A _F. Chapoton_, Feb 26 2018