A066165 Variant of Stanley's children's game. Class of n (named) children forms into rings of at least two with exactly one child inside each ring. a(n) gives number of possibilities, including clockwise order (or which hand is held), in each ring.

3, 8, 30, 234, 1680, 13040, 119448, 1212120, 13412520, 161968872, 2118607920, 29813747040, 449227822680, 7216747374720, 123128587713600, 2223511629522624, 42370586275466880, 849664985938704000, 17886165587251839360, 394366490810199895680, 9088843342633833461760

Offset: 3

Author

Len Smiley, Dec 12 2001

Keywords

• nonn
• nice
• easy

Examples

			a(4)=8: ring must have 3 of the four, fourth in middle. Two ways for the three to hold hands.
		

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.

Crossrefs

Cf. A066166 (original version).

Programs

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. *)

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 */

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