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A137775 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. 5
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; text; internal format)
OFFSET

0,3

COMMENTS

This sequence arises in a calculation of the fourth moments of the volumes of random polytopes in certain very symmetric convex bodies.

REFERENCES

M. Meckes, Volumens of symmetric random polytopes, Arch. Math. 82 (2004) 85--96.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

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. - Michael Somos, Jan 19 2011

G.f.: hypergeom([1,3],[],x/(1+3*x))/(1+3*x). - Mark van Hoeij, Nov 08 2011

a(n) ~ n! * exp(-3) * n^2/2. - Vaclav Kotesovec, Oct 08 2013

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

MATHEMATICA

Range[0, 20]! CoefficientList[Series[Exp[ -3x]/(1 - x)^3, {x, 0, 20}], x]

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 */

CROSSREFS

Cf. A000007, A000166, A024000, A087981.

Sequence in context: A088674 A203434 A076170 * A038050 A194080 A261841

Adjacent sequences:  A137772 A137773 A137774 * A137776 A137777 A137778

KEYWORD

nonn

AUTHOR

Mark W. Meckes (mark.meckes(AT)case.edu), May 06 2008

EXTENSIONS

Added a(0)=1 by Michael Somos, Jan 19 2011

STATUS

approved

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Last modified June 23 12:08 EDT 2021. Contains 345401 sequences. (Running on oeis4.)