A162973 Number of cycles in all derangement permutations of {1,2,...,n}.
0, 1, 2, 12, 64, 425, 3198, 27216, 258144, 2701737, 30933770, 384675148, 5163521856, 74417353985, 1146203362822, 18790377267840, 326682354342336, 6003886529652657, 116305541572943826, 2368629865508978284
Offset: 1
Keywords
Examples
a(4)=12 because in the derangements of {1,2,3,4}, namely (12)(34), (13)(24), (14)(23), (1234), (1243), (1324), (1342), (1423), and (1432), we have a total of 2+2+2+1+1+1+1+1+1=12 cycles.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..449
- John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
Programs
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Maple
G := exp(-z)*(z+ln(1-z))/(z-1): Gser := series(G, z = 0, 25): seq(factorial(n)*coeff(Gser, z, n), n = 1 .. 22);
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Mathematica
With[{nn=20},Rest[CoefficientList[Series[Exp[-x] (x+Log[1-x])/(x-1), {x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Jul 25 2013 *) -
PARI
x='x+O('x^30); concat([0], Vec(serlaplace(exp(-x)*(x+log(1-x))/(x -1)))) \\ G. C. Greubel, Sep 01 2018
Formula
a(n) = Sum_{k>=1} k*A008306(n,k).
E.g.f.: exp(-z)*(z+log(1-z))/(z-1).
a(n) ~ n! * (log(n) + gamma - 1)/exp(1), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 25 2013
D-finite with recurrence a(n) +2*(-n+2)*a(n-1) +(n-2)*(n-6)*a(n-2) +(3*n-8)*(n-3)*a(n-3) +3*(n-3)^2*a(n-4) +(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jul 26 2022