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

Author

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

Keywords

• nonn

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.

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.

Programs

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, tVaclav Kotesovec, Mar 12 2014 *)

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

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

Extensions

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