login
a(n) is the number of solutions to the congruence Sum_{k=1..n} x_k == 1 (mod 2n), where x_k are distinct elements of the set {0, 1, ..., 2n}, k = 1..n.
1

%I #12 Sep 05 2018 03:13:30

%S 1,4,18,192,3000,56160,1234800,32256000,979776000,33566400000,

%T 1279932192000,53950908211200,2490951541478400,124914111972249600,

%U 6761428395321600000,393000294670663680000,24412776290272161792000,1613964246117021646848000,113146793781167491817472000

%N a(n) is the number of solutions to the congruence Sum_{k=1..n} x_k == 1 (mod 2n), where x_k are distinct elements of the set {0, 1, ..., 2n}, k = 1..n.

%D V. S. Shevelev, On number of solutions of congruence Sum{i=1,...,s}x_i==r(modk), Izvestia Vuzov of the North-Caucasus region, Nature sciences, 2 (1997), 25-37 (in Russian).

%F a(n) = ((n-1)!/2)*Sum_{d|n} ( -1)^(n+d)*mu(n/d)*C(2d,d), where mu(n) is the Mobius function A008683.

%e If n=2, then we have the congruence x_1 + x_2 == 1 (mod 4), x_i is in {0,1,2,3}. Here we have 4 solutions: (0,1), (1,0), (2,3), (3,2); therefore a(2)=4.

%o (PARI) a(n) = ((n-1)!/2) * sumdiv(n, d, ( -1)^(n+d) * moebius(n/d) * binomial(2*d,d) );

%o vector(33,n,a(n)) \\ _Joerg Arndt_, Sep 05 2018

%Y Cf. A008683, A000984.

%K nonn

%O 1,2

%A _Vladimir Shevelev_, Mar 26 2010, Apr 09 2010, Jun 29 2010

%E a(6) corrected and more terms from _Joerg Arndt_, Sep 05 2018