OFFSET
3,1
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)
LINKS
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
FORMULA
E.g.f.: exp(-x*log(1-x)-x^2)-1.
a(n) = n!*sum(sum(binomial(k,j)*j!/(n-2*k+j)!*Stirling1(n-2*k+j,j)*(-1)^(n-k-j),j,0,k)/k!,k,1,floor(n/2)), n>2. - Vladimir Kruchinin, Sep 07 2010
a(n) ~ exp(-1) * n!. - Vaclav Kotesovec, Jun 04 2022
EXAMPLE
a(4)=8: ring must have 3 of the four, fourth in middle. Two ways for the three to hold hands.
MATHEMATICA
max = 20; f[x_] := Exp[-x*Log[1 - x] - x^2] - 1; Drop[ CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]!, 3] (* Jean-François Alcover, Oct 13 2011, after g.f. *)
PROG
(Maxima) a(n):=n!*sum(sum(binomial(k, j)*j!/(n-2*k+j)!*stirling1(n-2*k+j, j)*(-1)^(n-k-j), j, 0, k)/k!, k, 1, floor(n/2)); /* Vladimir Kruchinin, Sep 07 2010 */
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
Len Smiley, Dec 12 2001
STATUS
approved