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

3,1

References

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)

Links

Table of n, a(n) for n=3..23.

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

Cf. A066166 (original version).

Sequence in context: A066304 A298456 A145776 * A323775 A360605 A119838

Adjacent sequences: A066162 A066163 A066164 * A066166 A066167 A066168

Keyword

nonn,nice,easy

Author

Len Smiley, Dec 12 2001

Status

approved