A027710 Number of ways of placing n labeled balls into n unlabeled (but 3-colored) boxes.

1, 3, 12, 57, 309, 1866, 12351, 88563, 681870, 5597643, 48718569, 447428856, 4318854429, 43666895343, 461101962108, 5072054649573, 57986312752497, 687610920335610, 8442056059773267, 107135148331162767, 1403300026585387686, 18946012544520590991

Offset

0,2

Comments

Binomial transform of this sequence is A078940 and a(n+1) = 3*A078940(n). - Paul D. Hanna, Dec 08 2003

First column of the cube of the matrix exp(P)/exp(1) given in A011971. - Gottfried Helms, Mar 30 2007. Base matrix in A011971, second power in A078937, third power in A078938, fourth power in A078939.

The number of ways of putting n labeled balls into a set of bags and then putting the bags into 3 labeled boxes. - Peter Bala, Mar 23 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Amit Kumar Singh, Akash Kumar and Thambipillai Srikanthan, Accelerating Throughput-aware Run-time Mapping for Heterogeneous MPSoCs, ACM Transactions on Design Automation of Electronic Systems, 2012. - From N. J. A. Sloane, Dec 24 2012

Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.

Formula

E.g.f.: exp {3(e^x-1)}. - Michael Somos, Oct 18, 2002

a(n) = exp(-3)*Sum_{k>=0} 3^k*k^n/k!. - Benoit Cloitre, Sep 25 2003

G.f.: 3*(x/(1-x))*A(x/(1-x)) = A(x) - 1; thrice the binomial transform equals the sequence shifted one place left. - Paul D. Hanna, Dec 08 2003

a(n) = Sum_{k = 0..n} 3^k*A048993(n, k); A048993: Stirling2 numbers. - Philippe Deléham, May 09 2004

PE=exp(matpascal(5))/exp(1); A = PE^3; a(n)= A[ n,1 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^3; a(n)=A[ n,1]. - Gottfried Helms, Apr 08 2007

G.f.: (G(0) - 1)/(x-1)/3 where G(k) = 1 - 3/(1-k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013

G.f.: T(0)/(1-3*x), where T(k) = 1 - 3*x^2*(k+1)/( 3*x^2*(k+1) - (1-3*x-x*k)*(1-4*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 24 2013

a(n) ~ n^n * exp(n/LambertW(n/3)-3-n) / (sqrt(1+LambertW(n/3)) * LambertW(n/3)^n). - Vaclav Kotesovec, Mar 12 2014

G.f.: Sum_{j>=0} 3^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 07 2019

Maple

b:= proc(n, m) option remember; `if`(n=0,
      1, m*b(n-1, m)+3*b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Aug 03 2021

Mathematica

colors=3; Array[ bell, 25 ]; For[ x=1, x<=25, x++, bell[ x ]=0 ]; bell[ 1 ]=colors;
Print[ "1 ", colors ]; For[ n=2, n<=25, n++, bell[ n ]=colors*bell[ n-1 ];
For[ i=1, n-i>1, i++, bell[ n-i ]=bell[ n-i ]*(n-i)+colors*bell[ n-i-1 ] ];
bellsum=0; For[ t=0, t<n, t++, bellsum=bellsum+bell[ n-t ] ]; Print[ n, " ", bellsum ] ]
Table[BellB[n, 3], {n, 0, 20}] (* Vaclav Kotesovec, Mar 12 2014 *)

Prog

(PARI) a(n)=if(n<0, 0, n!*polcoeff(exp(3*(exp(x+x*O(x^n))-1)), n))
(Sage) from sage.combinat.expnums import expnums2
expnums(22, 3) # Zerinvary Lajos, Jun 26 2008

Crossrefs

Cf. A000110, A001861, A056857, A078937, A078938, A078940, A078944, A078945, A129323, A129324, A129325, A129327, A129328, A129329, A129331, A129332, A129333, A144180, A144223, A144263, A189233, A221159, A221176.

Sequence in context: A323631 A014333 A185618 * A307495 A302101 A279271

Adjacent sequences: A027707 A027708 A027709 * A027711 A027712 A027713

Keyword

nonn

Author

George Yuhasz (gyuhasz(AT)vt.edu) and John W. Layman

Extensions

Entry revised by N. J. A. Sloane, Apr 25 2007

Status

approved