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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. 2
3, 8, 30, 234, 1680, 13040, 119448, 1212120, 13412520, 161968872, 2118607920, 29813747040, 449227822680, 7216747374720, 123128587713600, 2223511629522624, 42370586275466880, 849664985938704000, 17886165587251839360, 394366490810199895680, 9088843342633833461760 (list; graph; refs; listen; history; text; internal format)
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
Cf. A066166 (original version).
Sequence in context: A066304 A298456 A145776 * A323775 A360605 A119838
KEYWORD
nonn,nice,easy
AUTHOR
Len Smiley, Dec 12 2001
STATUS
approved

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Last modified February 26 21:28 EST 2024. Contains 370352 sequences. (Running on oeis4.)