A161122 Number of cycles with entries of the same parity in all fixed-point-free involutions of {1,2,...,2n}.
0, 0, 2, 18, 180, 2100, 28350, 436590, 7567560, 145945800, 3101348250, 72020198250, 1814908995900, 49332526343100, 1438865351673750, 44826189802143750, 1485668004871050000, 52196469237802890000, 1937793920453432291250, 75801938653031321981250, 3116301922402398792562500
Offset: 0
Keywords
Examples
a(2)=2 because in the 3 permutations (12)(34), (13)(24), (14)(23) we have a total of 2 cycles with entries of the same parity.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..300
Programs
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Magma
DoubleFactorial:=func< n | &*[n..2 by -2] >; [ n*(n-1)*DoubleFactorial(2*n-3): n in [0..22]]; // Vincenzo Librandi, Jul 21 2017
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Maple
seq(n*(n-1)*(product(2*j-1, j = 1 .. n-1)), n = 0 .. 18);
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Mathematica
Table[n (n - 1) (2 n -3)!!, {n, 0, 20}] (* Vincenzo Librandi, Jul 21 2017 *)
Formula
a(n) = n(n-1)(2n-3)!!.
a(n) = Sum_{k>=0} k*A161121(n,k).
D-finite with recurrence (-n+2)*a(n) +n*(2*n-3)*a(n-1)=0. - R. J. Mathar, Jul 26 2022