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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. 5

%I #53 Feb 17 2021 02:55:57

%S 1,2,32,945,40992,2350950,167829629,14342726398,1427875921472,

%T 162295947266310,20738354463124740,2942918038945276392,

%U 459208250931426639151,78145305037982571857910,14403186440935002502579620,2858375634375573872689073400,607685050482829924986457079520

%N 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 Alois P. Heinz, <a href="/A321712/b321712.txt">Table of n, a(n) for n = 0..311</a>

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

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

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

%F a(n) = ((2*n)!/n!) * [x^(2*n)] (exp(exp(x) - 1) - 1)^n. - _Ilya Gutkovskiy_, Feb 15 2021

%F From _Vaclav Kotesovec_, Feb 17 2021: (Start)

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

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

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

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

%F r = 0.49039351286814033601311908705923238442641817550970055325385921966197159992...

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

%F and c = 0.1809999195056310772963776575864895285358912769365095026676184958683437... (End)

%p b:= proc(n, m, k) option remember; `if`(n=0, 1, add(

%p b(n-1, max(j, m), k)*`if`(j>m, k, 1) , j=1..m+1))

%p end:

%p a:= n-> add(b(2*n, 0, n-i)*(-1)^i*binomial(n, i), i=0..n)/n!:

%p seq(a(n), n=0..15);

%t 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 a[n_] := Sum[b[2n, 0, n - i] (-1)^i Binomial[n, i], {i, 0, n}]/n!;

%t a /@ Range[0, 15] (* _Jean-François Alcover_, Dec 08 2020, after _Alois P. Heinz_ *)

%t Table[Sum[StirlingS2[2*n, k] * StirlingS2[k, n], {k, n, 2*n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Feb 17 2021 *)

%Y Cf. A000110, A008277, A039810, A130191.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Aug 27 2019

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Last modified March 29 04:59 EDT 2024. Contains 371264 sequences. (Running on oeis4.)