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

%I A027710 #65 Jul 04 2025 17:25:33
%S A027710 1,3,12,57,309,1866,12351,88563,681870,5597643,48718569,447428856,
%T A027710 4318854429,43666895343,461101962108,5072054649573,57986312752497,
%U A027710 687610920335610,8442056059773267,107135148331162767,1403300026585387686,18946012544520590991
%N A027710 Number of ways of placing n labeled balls into n unlabeled (but 3-colored) boxes.
%C A027710 Binomial transform of this sequence is A078940 and a(n+1) = 3*A078940(n). - _Paul D. Hanna_, Dec 08 2003
%C A027710 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 A027710 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 A027710 Vincenzo Librandi, <a href="/A027710/b027710.txt">Table of n, a(n) for n = 0..200</a>
%H A027710 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
%H A027710 Jacob Sprittulla, <a href="https://arxiv.org/abs/2008.09984">On Colored Factorizations</a>, arXiv:2008.09984 [math.CO], 2020.
%F A027710 E.g.f.: exp {3(e^x-1)}. - _Michael Somos_, Oct 18 2002
%F A027710 a(n) = exp(-3)*Sum_{k>=0} 3^k*k^n/k!. - _Benoit Cloitre_, Sep 25 2003
%F A027710 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 A027710 a(n) = Sum_{k = 0..n} 3^k*A048993(n, k); A048993: Stirling2 numbers. - _Philippe Deléham_, May 09 2004
%F A027710 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 A027710 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 A027710 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 A027710 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 A027710 G.f.: Sum_{j>=0} 3^j*x^j / Product_{k=1..j} (1 - k*x). - _Ilya Gutkovskiy_, Apr 07 2019
%p A027710 b:= proc(n, m) option remember; `if`(n=0,
%p A027710       1, m*b(n-1, m)+3*b(n-1, m+1))
%p A027710     end:
%p A027710 a:= n-> b(n, 0):
%p A027710 seq(a(n), n=0..27);  # _Alois P. Heinz_, Aug 03 2021
%t A027710 colors=3; Array[ bell, 25 ]; For[ x=1, x<=25, x++, bell[ x ]=0 ]; bell[ 1 ]=colors;
%t A027710 Print[ "1 ", colors ]; For[ n=2, n<=25, n++, bell[ n ]=colors*bell[ n-1 ];
%t A027710 For[ i=1, n-i>1, i++, bell[ n-i ]=bell[ n-i ]*(n-i)+colors*bell[ n-i-1 ] ];
%t A027710 bellsum=0; For[ t=0, t<n, t++, bellsum=bellsum+bell[ n-t ] ]; Print[ n, " ", bellsum ] ]
%t A027710 Table[BellB[n,3],{n,0,20}] (* _Vaclav Kotesovec_, Mar 12 2014 *)
%o A027710 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(3*(exp(x+x*O(x^n))-1)),n))
%o A027710 (Sage) from sage.combinat.expnums import expnums2
%o A027710 expnums(22, 3) # _Zerinvary Lajos_, Jun 26 2008
%Y A027710 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 A027710 nonn
%O A027710 0,2
%A A027710 George Yuhasz (gyuhasz(AT)vt.edu) and _John W. Layman_
%E A027710 Entry revised by _N. J. A. Sloane_, Apr 25 2007