login
A174668
a(n) is the number of solutions of the congruence Sum_{k=1..n} x_k == n (mod 2n) where x_k are distinct elements of the set {0,1,...,2n}.
0
1, 2, 24, 216, 3120, 54720, 1239840, 32618880, 981227520, 33479308800, 1279972108800, 53991144345600, 2490957768499200, 124892840469196800, 6761466317878272000, 393017221207683072000, 24412776645959589888000, 1613947446288417816576000, 113146793902812592226304000
OFFSET
1,2
REFERENCES
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).
FORMULA
a(n) = ((n-1)!/2)*Sum_{d|n} (-1)^(n+d)*phi(n/d)*C(2d,d), where phi(n) is the Euler totient function A000010 (it is of interest that our formulas for A174663(n) and for a(n) differ only in the factor in the summands: mu(n/d) or phi(n/d)).
EXAMPLE
If n=2, then we have the congruence x_1 + x_2 == 2 (mod 4), x_i is in {0,1,2,3}. Here we have only two solutions: (0,2) and (2,0) in the condition x_1<(>)x_2.
PROG
(PARI) a(n) = ((n-1)!/2) * sumdiv(n, d, ( -1)^(n+d) * eulerphi(n/d) * binomial(2*d, d) );
vector(33, n, a(n)) \\ Joerg Arndt, Sep 05 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Mar 26 2010, Jun 29 2010
EXTENSIONS
a(9) corrected and more terms from Joerg Arndt, Sep 05 2018
STATUS
approved