A144180 Number of ways of placing n labeled balls into n unlabeled (but 5-colored) boxes.

1, 5, 30, 205, 1555, 12880, 115155, 1101705, 11202680, 120415755, 1362057155, 16151603830, 200144023805, 2584429030505, 34691478901030, 483040313859705, 6963313750468055, 103747357497925880, 1595132080103893655

Offset: 0

Author

Philippe Deléham, Sep 12 2008

Keywords

• nonn

Comments

a(n) is also the exp transform of A010716. - 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 5 labeled boxes. - Peter Bala, Mar 23 2013

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013; Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000110, A001861, A027710, A078944. A144223, 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))*5)
    end:
seq(a(n), n=0..25); # Alois P. Heinz, Oct 09 2008

Mathematica

Table[BellB[n,5],{n,0,20}] (* Vaclav Kotesovec, Mar 12 2014 *)

Sage

expnums(19, 5) # Zerinvary Lajos, May 15 2009

Formula

a(n) = Sum_{k=0..n} 5^k * A048993(n,k); A048993: Stirling2 numbers.

G.f.: A(x) satisfies 5*(x/(1-x))*A(x/(1-x)) = A(x)-1; five times the binomial transform equals this sequence shifted one place left.

E.g.f.: exp(5*(exp(x)-1)).

G.f.: (G(0) - 1)/(x-1)/5 where G(k) = 1 - 5/(1-k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013

a(n) ~ n^n * exp(n/LambertW(n/5)-5-n) / (sqrt(1+LambertW(n/5)) * LambertW(n/5)^n). - Vaclav Kotesovec, Mar 12 2014

G.f.: Sum_{j>=0} 5^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