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A066166
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Stanley's children's game. Class of n (named) children forms into rings with exactly one child inside each ring. We allow the case when outer ring has only one child. a(n) gives number of possibilities, including clockwise order (or which hand is held), in each ring.
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25
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2, 3, 20, 90, 594, 4200, 34544, 316008, 3207240, 35699400, 432690312, 5672581200, 79991160144, 1207367605080, 19423062612480, 331770360922560, 5997105160795584, 114373526841360000, 2295170834453089920
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OFFSET
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2,1
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COMMENTS
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Apparently n divides a(n), so a(n)/n = 1, 1, 5, 18, 99, 600, 4318, 35112, 320724, 3245400, 36057526, 436352400, 5713654296, ... - R. J. Mathar, Oct 31 2015
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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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FORMULA
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E.g.f.: -1+1/(1-x)^x.
a(n) ~ n! * (1 - 1/n + (1-log(n)-gamma)/n^2), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Apr 21 2014
a(n) = b(n), n>0, a(0)=0, where b(n) = (n-1)!*Sum_{i=1..n-1} (1+1/i)*b(n-i-1)/(n-i-1)!, b(0)=1. - Vladimir Kruchinin, Feb 25 2015
E.g.f.: Sum_{n>=1} x^n/n! * Product_{k=0..n-1} (k + x). - Paul D. Hanna, Oct 26 2015
a(n) = n! * Sum_{k=0..floor(n/2)} |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, May 10 2022
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EXAMPLE
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a(4)=20: 12 ways to make 2 hugs, 8 ways to make a 3-ring.
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MATHEMATICA
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Drop[With[{nn=20}, CoefficientList[Series[1/(1-x)^x-1, {x, 0, nn}], x] Range[ 0, nn]!], 2] (* Harvey P. Dale, Sep 17 2011 *)
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PROG
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(PARI) a(n)=if(n<0, 0, n!*polcoeff(-1+1/(1-x+x*O(x^n))^x, n))
(PARI) {a(n) = n!*polcoeff( sum(m=1, n, x^m/m! * prod(k=0, m-1, x + k) +x*O(x^n) ), n)}
(PARI) a(n) = n!*sum(k=0, n\2, abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, May 10 2022
(Maxima)
b(n):=if n=0 then 1 else (n-1)!*sum((1+1/i)*b(n-i-1)/(n-i-1)!, i, 1, n-1);
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(-1+1/(1-x)^x)); [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Aug 29 2018
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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