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A027710 Number of ways of placing n labeled balls into n unlabeled (but 3-colored) boxes. 27

%I

%S 1,3,12,57,309,1866,12351,88563,681870,5597643,48718569,447428856,

%T 4318854429,43666895343,461101962108,5072054649573,57986312752497,

%U 687610920335610,8442056059773267,107135148331162767

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

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

%C 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.

%C 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

%H Vincenzo Librandi, <a href="/A027710/b027710.txt">Table of n, a(n) for n = 0..200</a>

%H Amit Kumar Singh, Akash Kumar and Thambipillai Srikanthan, <a href="http://www.ece.nus.edu.sg/stfpage/eleak/pdf/akumar_todaes_2012.pdf">Accelerating Throughput-aware Run-time Mapping for Heterogeneous MPSoCs</a>, ACM Transactions on Design Automation of Electronic Systems, 2012. - From _N. J. A. Sloane_, Dec 24 2012

%F E.g.f.: exp {3(e^x-1)}. - _Michael Somos_, Oct 18, 2002

%F a(n) = exp(-3)*sum(k>=0, 3^k*k^n/k! ) - _Benoit Cloitre_, Sep 25 2003

%F 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

%F a(n) = Sum_{k = 0..n} 3^k*A048993(n, k); A048993: Stirling-2 numbers . - _Philippe Deléham_, May 09 2004

%F 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

%F 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

%F 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

%F 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

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

%t colors=3; Array[ bell, 25 ]; For[ x=1, x<=25, x++, bell[ x ]=0 ]; bell[ 1 ]=colors;

%t Print[ "1 ", colors ]; For[ n=2, n<=25, n++, bell[ n ]=colors*bell[ n-1 ];

%t For[ i=1, n-i>1, i++, bell[ n-i ]=bell[ n-i ]*(n-i)+colors*bell[ n-i-1 ] ];

%t bellsum=0; For[ t=0, t<n, t++, bellsum=bellsum+bell[ n-t ] ]; Print[ n, " ", bellsum ] ]

%t Table[BellB[n,3],{n,0,20}] (* _Vaclav Kotesovec_, Mar 12 2014 *)

%o (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(3*(exp(x+x*O(x^n))-1)),n))

%o (Sage) from sage.combinat.expnums import expnums2

%o expnums(22, 3) # _Zerinvary Lajos_, Jun 26 2008

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

%K nonn

%O 0,2

%A George Yuhasz (gyuhasz(AT)vt.edu) and _John W. Layman_

%E Entry revised by _N. J. A. Sloane_, Apr 25 2007

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Last modified November 17 13:33 EST 2019. Contains 329230 sequences. (Running on oeis4.)