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A137775
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Number of triples of permutations on n letters such that for each j, exactly one of the permutations fixes j and the other two have the same image on j.
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2
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1, 0, 3, 6, 45, 252, 1935, 16146, 153657, 1616760, 18699579, 235498590, 3207570597, 46968796404, 735689606535, 12272343940458, 217191191400945, 4064131571557104, 80166987477918963, 1662468879466624950
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| This sequence arises in a calculation of the fourth moments of the volumes of random polytopes in certain very symmetric convex bodies.
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REFERENCES
| M. Meckes, Volumens of symmetric random polytopes, Arch. Math. 82 (2004) 85--96.
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FORMULA
| a(n) = n(a(n-1)+3*a(n-2)) with a(0)=1; E.g.f: exp(-3x)/(1-x)^3.
a(n) is the number of derangements (permutations with no fixed points) of n elements where each cycle is colored with one of three colors. [From Michael Somos, Jan 19 2011]
G.f.: hypergeom([1,3],[],x/(1+3*x))/(1+3*x) - Mark van Hoeij, Nov 08 2011
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EXAMPLE
| a(2)=3 because one of the permutations must be the identity and the other two are the transposition (1 2); there are three ways to pick which is the identity.
a(4) = 45 because there are 6 derangements with one 4-cycle with 3^1 ways to color each derangement and 3 derangements with two 2-cycles with 3^2 ways to color each derangement. - Michael Somos, Jan 19 2011
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MATHEMATICA
| Range[0, 20]! CoefficientList[Series[Exp[ -3x]/(1 - x)^3, {x, 0, 20}], x]
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PROG
| (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( -3 * x + x * O(x^n) ) / ( 1 - x )^3, n ) )} /* Michael Somos, Jan 19 2011 */
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CROSSREFS
| Cf. A000007, A000166, A024000, A087981.
Sequence in context: A088674 A203434 A076170 * A038050 A194080 A197884
Adjacent sequences: A137772 A137773 A137774 * A137776 A137777 A137778
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KEYWORD
| nonn
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AUTHOR
| Mark W. Meckes (mark.meckes(AT)case.edu), May 06 2008
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EXTENSIONS
| Added a(0)=1 by Michael Somos, Jan 19 2011
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