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.

1, 2, 32, 945, 40992, 2350950, 167829629, 14342726398, 1427875921472, 162295947266310, 20738354463124740, 2942918038945276392, 459208250931426639151, 78145305037982571857910, 14403186440935002502579620, 2858375634375573872689073400, 607685050482829924986457079520

Offset: 0

Author

Alois P. Heinz, Aug 27 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..311
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A008277, A039810, A130191.

Programs

Maple

b:= proc(n, m, k) option remember; `if`(n=0, 1, add(
      b(n-1, max(j, m), k)*`if`(j>m, k, 1) , j=1..m+1))
    end:
a:= n-> add(b(2*n, 0, n-i)*(-1)^i*binomial(n, i), i=0..n)/n!:
seq(a(n), n=0..15);

Mathematica

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}]];
a[n_] := Sum[b[2n, 0, n - i] (-1)^i Binomial[n, i], {i, 0, n}]/n!;
a /@ Range[0, 15] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)
Table[Sum[StirlingS2[2*n, k] * StirlingS2[k, n], {k, n, 2*n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 17 2021 *)

Formula

a(n) = Sum_{i=n..2*n} Stirling2(2*n,i)*Stirling2(i,n).

a(n) = A039810(2n,n) = A130191(2n,n).

a(n) = ((2*n)!/n!) * [x^(2*n)] (exp(exp(x) - 1) - 1)^n. - Ilya Gutkovskiy, Feb 15 2021

From Vaclav Kotesovec, Feb 17 2021: (Start)

a(n) ~ c * d^n * (n-1)!, where

d = -4/(p^2*q*(1 + q + r)) = 14.158467948361614323478778011058425244554144983745335637776404207122781371002...

p = LambertW(-2/((1+r)*exp(2/(1+r))))

q = LambertW(-(1+r)/exp(1+r))

r = 0.49039351286814033601311908705923238442641817550970055325385921966197159992...

is the root of the equation p*(1+r)*(1+q+r) + (2 + p + p*r) = 0

and c = 0.1809999195056310772963776575864895285358912769365095026676184958683437... (End)