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

1, 6, 42, 330, 2850, 26682, 268098, 2869242, 32510850, 388109562, 4861622850, 63682081530, 869725707522, 12352785293562, 182049635623362, 2778394592545530, 43833623157604482, 713738052924821754

Offset: 0

Author

Philippe Deléham, Sep 14 2008

Keywords

• nonn

Comments

a(n) is also the exp transform of A010722. - Alois P. Heinz, Oct 09 2008

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• N. J. A. Sloane, Transforms

Crossrefs

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

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*6)
    end:
seq(a(n), n=0..25); # Alois P. Heinz, Oct 09 2008

Mathematica

Table[BellB[n,6],{n,0,20}] (* Vaclav Kotesovec, Mar 12 2014 *)

Sage

expnums(18, 6) # Zerinvary Lajos, May 15 2009

Formula

a(n) = Sum_{k=0..n} 6^k*A048993(n,k); A048993: Stirling2 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)).

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

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

G.f.: Sum_{j>=0} 6^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 07 2019

Extensions

More terms from Alois P. Heinz, Oct 09 2008