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A332244 Number of chains of length n in the lattice of set partitions of [2n] ordered by refinement.

%I A332244 #17 Jul 03 2025 07:57:17
%S A332244 1,2,45,8176,5967927,12354550875,58745934381618,557269710685272585,
%T A332244 9536970947556120868800,273107814151944184186060560,
%U A332244 12345107536247705318429028256740,840776466106445249176017830108333910,83061829400676435859326576547506817501212
%N A332244 Number of chains of length n in the lattice of set partitions of [2n] ordered by refinement.
%H A332244 Alois P. Heinz, <a href="/A332244/b332244.txt">Table of n, a(n) for n = 0..133</a>
%H A332244 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_(order)">Lattice (order)</a>
%H A332244 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A332244 a(n) = A331955(2n,n).
%F A332244 Limit_{n->oo} (a(n)/n!^4)^(1/n) = 4/(w*(2-w)) = 6.17655460948348035823168..., where w = -LambertW(-2*exp(-2)) = -A226775. - _Vaclav Kotesovec_, Jul 03 2025
%p A332244 b:= proc(n, k, t) option remember; `if`(k<0, 0, `if`({n, k}={0}, 1,
%p A332244       add(`if`(k=1, 1, b(v, k-1, 1))*Stirling2(n, v), v=k..n-t)))
%p A332244     end:
%p A332244 a:= n-> b(2*n, n, 0):
%p A332244 seq(a(n), n=0..14);
%t A332244 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 A332244 a[n_] := b[2n, n, 0];
%t A332244 a /@ Range[0, 14] (* _Jean-François Alcover_, May 08 2020, after Maple *)
%Y A332244 Cf. A331955.
%K A332244 nonn
%O A332244 0,2
%A A332244 _Alois P. Heinz_, Feb 07 2020