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A321712 Number of partitions of a 2n-set into colored blocks, such that exactly n colors are used and the colors are introduced in increasing order.

%I A321712 #53 Feb 17 2021 02:55:57
%S A321712 1,2,32,945,40992,2350950,167829629,14342726398,1427875921472,
%T A321712 162295947266310,20738354463124740,2942918038945276392,
%U A321712 459208250931426639151,78145305037982571857910,14403186440935002502579620,2858375634375573872689073400,607685050482829924986457079520
%N A321712 Number of partitions of a 2n-set into colored blocks, such that exactly n colors are used and the colors are introduced in increasing order.
%H A321712 Alois P. Heinz, <a href="/A321712/b321712.txt">Table of n, a(n) for n = 0..311</a>
%H A321712 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A321712 a(n) = Sum_{i=n..2*n} Stirling2(2*n,i)*Stirling2(i,n).
%F A321712 a(n) = A039810(2n,n) = A130191(2n,n).
%F A321712 a(n) = ((2*n)!/n!) * [x^(2*n)] (exp(exp(x) - 1) - 1)^n. - _Ilya Gutkovskiy_, Feb 15 2021
%F A321712 From _Vaclav Kotesovec_, Feb 17 2021: (Start)
%F A321712 a(n) ~ c * d^n * (n-1)!, where
%F A321712 d = -4/(p^2*q*(1 + q + r)) = 14.158467948361614323478778011058425244554144983745335637776404207122781371002...
%F A321712 p = LambertW(-2/((1+r)*exp(2/(1+r))))
%F A321712 q = LambertW(-(1+r)/exp(1+r))
%F A321712 r = 0.49039351286814033601311908705923238442641817550970055325385921966197159992...
%F A321712 is the root of the equation p*(1+r)*(1+q+r) + (2 + p + p*r) = 0
%F A321712 and c = 0.1809999195056310772963776575864895285358912769365095026676184958683437... (End)
%p A321712 b:= proc(n, m, k) option remember; `if`(n=0, 1, add(
%p A321712       b(n-1, max(j, m), k)*`if`(j>m, k, 1) , j=1..m+1))
%p A321712     end:
%p A321712 a:= n-> add(b(2*n, 0, n-i)*(-1)^i*binomial(n, i), i=0..n)/n!:
%p A321712 seq(a(n), n=0..15);
%t A321712 b[n_, m_, k_] := b[n, m, k] = If[n == 0, 1, Sum[b[n - 1, Max[j, m], k] If[j > m, k, 1] , {j, 1, m + 1}]];
%t A321712 a[n_] := Sum[b[2n, 0, n - i] (-1)^i Binomial[n, i], {i, 0, n}]/n!;
%t A321712 a /@ Range[0, 15] (* _Jean-François Alcover_, Dec 08 2020, after _Alois P. Heinz_ *)
%t A321712 Table[Sum[StirlingS2[2*n, k] * StirlingS2[k, n], {k, n, 2*n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Feb 17 2021 *)
%Y A321712 Cf. A000110, A008277, A039810, A130191.
%K A321712 nonn
%O A321712 0,2
%A A321712 _Alois P. Heinz_, Aug 27 2019