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A066165
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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.
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1
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3, 8, 30, 234, 1680, 13040, 119448, 1212120, 13412520, 161968872, 2118607920, 29813747040, 449227822680, 7216747374720, 123128587713600, 2223511629522624, 42370586275466880, 849664985938704000
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OFFSET
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3,1
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)
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LINKS
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Table of n, a(n) for n=3..20.
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FORMULA
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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
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EXAMPLE
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a(4)=8: ring must have 3 of the four, fourth in middle. Two ways for the three to hold hands.
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MATHEMATICA
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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. *)
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PROG
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(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 */
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CROSSREFS
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Cf. A066166 (original version).
Sequence in context: A066304 A298456 A145776 * A323775 A119838 A148889
Adjacent sequences: A066162 A066163 A066164 * A066166 A066167 A066168
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Len Smiley, Dec 12 2001
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STATUS
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approved
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