0,2

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.

Zerinvary Lajos, Sage Notebooks

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

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 ] ]

(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

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: A117107 A128326 A014333 this_sequence A009248 A012709 A032268

Adjacent sequences: A027707 A027708 A027709 this_sequence A027711 A027712 A027713

nonn

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

Entry revised by njas, Apr 25 2007