OFFSET
3,1
COMMENTS
The reference to A094047 arises in the formula because that sequence is also the number of directed Hamiltonian cycles in the n-crown graph. (Each cycle can be broken in 2n ways to give a path.) - Andrew Howroyd, Feb 21 2016
Also, the number of ways of seating n married couples at 2*n chairs arranged side-by-side in a straight line, men and women in alternate positions, so that no husband is next to his wife. - Andrew Howroyd, Sep 19 2017
LINKS
Seiichi Manyama, Table of n, a(n) for n = 3..253 (terms 3..50 from Andrew Howroyd)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems
Eric Weisstein's World of Mathematics, Crown Graph
Eric Weisstein's World of Mathematics, Hamiltonian Path
FORMULA
a(n) ~ 4*Pi*n^(2*n+1) / exp(2*n+2). - Vaclav Kotesovec, Feb 25 2016
a(n) = (n-1)*n*a(n-1) + (n-1)^2*n*a(n-2) + (n-2)*(n-1)*n*a(n-3). - Vaclav Kotesovec, Feb 25 2016
a(n) = 2*n! * A000271(n). - Andrew Howroyd, Sep 19 2017
MATHEMATICA
Table[2 n! Sum[(-1)^(n - k) k! Binomial[n + k, 2 k], {k, 0, n}], {n, 3, 20}] (* Eric W. Weisstein, Sep 20 2017 *)
Table[2 (-1)^n n! HypergeometricPFQ[{1, -n, n + 1}, {1/2}, 1/4], {n, 3, 20}] (* Eric W. Weisstein, Sep 20 2017 *)
PROG
(PARI) /* needs the routine nhp() from the Alekseyev link */
{ A137886(n) = nhp( matrix(2*n, 2*n, i, j, if(min(i, j)<=n && max(i, j)>n && abs(j-i)!=n, 1, 0)) ) }
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Feb 20 2008
EXTENSIONS
More terms from Max Alekseyev, Feb 13 2009
a(14) from Eric W. Weisstein, Jan 15 2014
a(15)-a(16) from Andrew Howroyd, Feb 21 2016
STATUS
approved