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

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..311

Wikipedia, Partition of a set

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)

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 *)

Crossrefs

Cf. A000110, A008277, A039810, A130191.

Sequence in context: A069182 A333772 A012233 * A211488 A012119 A282835

Adjacent sequences: A321709 A321710 A321711 * A321713 A321714 A321715

Keyword

nonn

Author

Alois P. Heinz, Aug 27 2019

Status

approved