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A370253 Number of deranged matchings of 2n people with partners (of either sex) such that at least one person is matched with their spouse.

%I A370253 #46 Feb 29 2024 11:29:33
%S A370253 0,1,1,7,45,401,4355,56127,836353,14144545,267629139,5601014255,
%T A370253 128455425593,3203605245777,86317343312395,2498680706048191,
%U A370253 77336483434140705,2548534969132415297,89087730603300393443,3292572900736818264015,128281460895447809211529
%N A370253 Number of deranged matchings of 2n people with partners (of either sex) such that at least one person is matched with their spouse.
%H A370253 Alois P. Heinz, <a href="/A370253/b370253.txt">Table of n, a(n) for n = 0..404</a>
%F A370253 a(n) = A001147(n) - A053871(n).
%F A370253 a(n) = Sum_{i=0..n-1} (-1)^(n - i + 1) * binomial(n,i)*A001147(i).
%F A370253 a(n) mod 2 = A057427(n).
%F A370253 a(n) = Sum_{k=1..n} A055140(n,k). - _Alois P. Heinz_, Feb 14 2024
%e A370253 For n=0, there is no matching which has at least one person matched with their original partner.
%e A370253 For n=1, there are only 2 people, so there is only one way to match them and it is with their original partner.
%e A370253 For n=2, we have two couples, A0 with A1, and B0 with B1. Of the three ways to match them [(A0,A1),(B0,B1)], [(A0,B0),(A1,B1)] and [(A0,B1),(A1,B0)], only the first matching has a person matched up with their original partner.
%p A370253 a:= proc(n) option remember; `if`(n<3, signum(n),
%p A370253       (4*n-7)*a(n-1)-2*(2*n^2-10*n+11)*a(n-2)-2*(n-2)*(2*n-5)*a(n-3))
%p A370253     end:
%p A370253 seq(a(n), n=0..20);  # _Alois P. Heinz_, Feb 14 2024
%t A370253 a[n_] := Sum[(-1)^(n-i+1)*Binomial[n, i]*(2i-1)!!, {i, 0, n-1}];
%t A370253 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 29 2024 *)
%o A370253 (Python)
%o A370253 import math
%o A370253 A001147 = lambda i: math.factorial(2*i) // ( 2 ** i * math.factorial(i) )
%o A370253 A370253 = lambda n: int( sum( (-1)**(i+1) * math.comb(n,n-i) * A001147(n-i) for i in range(1,n+1) ) )
%o A370253 print( ", ".join( str(A370253(i)) for i in range(0,21) ) )
%Y A370253 Cf. A001147 (total number of matchings for 2n people).
%Y A370253 Cf. A053871 (number of deranged matchings of 2n people with partners (of either sex) other than their spouse).
%Y A370253 Cf. A055140, A057427.
%K A370253 nonn
%O A370253 0,4
%A A370253 _Sam Coutteau_, Feb 13 2024