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A027710
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Number of ways of placing n labeled balls into n unlabeled (but 3-colored) boxes.
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19
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1, 3, 12, 57, 309, 1866, 12351, 88563, 681870, 5597643, 48718569, 447428856, 4318854429, 43666895343, 461101962108, 5072054649573, 57986312752497, 687610920335610, 8442056059773267, 107135148331162767
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Binomial transform of this sequence is A078940 and a(n+1) = 3*A078940(n). - Paul D. Hanna (pauldhanna(AT)juno.com), Dec 08 2003
First column of the cube of the matrix exp(P)/exp(1) given in A011971. - Gottfried Helms (helms(AT)uni-kassel.de), Mar 30 2007. Base matrix in A011971, second power in A078937, third power in A078938, fourth power in A078939.
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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 (benoit7848c(AT)orange.fr), 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 (pauldhanna(AT)juno.com), Dec 08 2003
a(n) = Sum_{k = 0..n} 3^k*A048993(n, k); A048993: Stirling-2 numbers . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 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 (helms(AT)uni-kassel.de), Apr 08 2007
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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 ] ]
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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 sage: expnums(22, 3) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008
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CROSSREFS
| Cf. A000110, A001861, A078940.
Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.
Sequence in context: A128326 A014333 A185618 * A009248 A012709 A032268
Adjacent sequences: A027707 A027708 A027709 * A027711 A027712 A027713
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KEYWORD
| nonn
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AUTHOR
| George Yuhasz (gyuhasz(AT)vt.edu) and John Layman (layman(AT)math.vt.edu)
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EXTENSIONS
| Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Apr 25 2007
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