A332244 - OEIS

%I #15 May 08 2020 16:36:25

%S 1,2,45,8176,5967927,12354550875,58745934381618,557269710685272585,

%T 9536970947556120868800,273107814151944184186060560,

%U 12345107536247705318429028256740,840776466106445249176017830108333910,83061829400676435859326576547506817501212

%N Number of chains of length n in the lattice of set partitions of [2n] ordered by refinement.

%H Alois P. Heinz, <a href="/A332244/b332244.txt">Table of n, a(n) for n = 0..133</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_(order)">Lattice (order)</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F a(n) = A331955(2n,n).

%p b:= proc(n, k, t) option remember; `if`(k<0, 0, `if`({n, k}={0}, 1,

%p       add(`if`(k=1, 1, b(v, k-1, 1))*Stirling2(n, v), v=k..n-t)))

%p     end:

%p a:= n-> b(2*n, n, 0):

%p seq(a(n), n=0..14);

%t b[n_, k_, t_] := b[n, k, t] = If[k < 0, 0, If[Union@{n, k} == {0}, 1, Sum[ If[k == 1, 1, b[v, k - 1, 1]] StirlingS2[n, v], {v, k, n - t}]]];

%t a[n_] := b[2n, n, 0];

%t a /@ Range[0, 14] (* _Jean-François Alcover_, May 08 2020, after Maple *)

%Y Cf. A331955.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Feb 07 2020