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A144223 Number of ways of placing n labeled balls into n unlabeled (but 6-colored) boxes. 14

%I

%S 1,6,42,330,2850,26682,268098,2869242,32510850,388109562,4861622850,

%T 63682081530,869725707522,12352785293562,182049635623362,

%U 2778394592545530,43833623157604482,713738052924821754

%N Number of ways of placing n labeled balls into n unlabeled (but 6-colored) boxes.

%C a(n) is also the exp transform of A010722. - _Alois P. Heinz_, Oct 09 2008

%C The number of ways of putting n labeled balls into a set of bags and then putting the bags into 6 labeled boxes. - _Peter Bala_, Mar 23 2013

%H Alois P. Heinz, <a href="/A144223/b144223.txt">Table of n, a(n) for n = 0..200</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F a(n) = Sum_{k=0..n} 6^k*A048993(n,k); A048993: Stirling-2 numbers. G.f.: 6*(x/(1-x))*A(x/(1-x))=A(x)-1; six times the binomial transform equals this sequence shifted one place left. E.g.f.: exp(6(e^x-1)).

%F G.f.: T(0)/(1-6*x), where T(k) = 1 - 6*x^2*(k+1)/(6*x^2*(k+1) - (1-6*x-x*k)*(1-7*x-x*k)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Dec 04 2013

%F a(n) ~ n^n * exp(n/LambertW(n/6)-6-n) / (sqrt(1+LambertW(n/6)) * LambertW(n/6)^n). - _Vaclav Kotesovec_, Mar 12 2014

%p a:= proc(n) option remember; `if`(n=0, 1,

%p (1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*6)

%p end:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Oct 09 2008

%t Table[BellB[n,6],{n,0,20}] (* _Vaclav Kotesovec_, Mar 12 2014 *)

%o (Sage) expnums(18, 6)# [_Zerinvary Lajos_, May 15 2009]

%Y Cf. A000110, A001861, A027710, A078944, A144180. A144263, A189233, A221159, A221176.

%K nonn

%O 0,2

%A _Philippe Deléham_, Sep 14 2008

%E More terms from _Alois P. Heinz_, Oct 09 2008

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Last modified March 20 13:18 EDT 2019. Contains 321345 sequences. (Running on oeis4.)