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}.

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).

Links

Table of n, a(n) for n=1..19.

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

Cf. A174663, A000010, A008683, A000984.

Sequence in context: A245019 A189769 A208533 * A302444 A121213 A147538

Adjacent sequences: A174665 A174666 A174667 * A174669 A174670 A174671

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