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A066166
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.
25
2, 3, 20, 90, 594, 4200, 34544, 316008, 3207240, 35699400, 432690312, 5672581200, 79991160144, 1207367605080, 19423062612480, 331770360922560, 5997105160795584, 114373526841360000, 2295170834453089920
OFFSET
2,1
COMMENTS
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
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.
P. Flajolet, S. Gerhold and B. Salvy, On the non-holonomic character of logarithms, powers and the n-th prime function, arXiv:math/0501379 [math.CO], 2005.
FORMULA
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
EXAMPLE
a(4)=20: 12 ways to make 2 hugs, 8 ways to make a 3-ring.
MATHEMATICA
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 *)
PROG
(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)}
for(n=2, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 26 2015
(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);
makelist(a(n), n, 2, 10); /* Vladimir Kruchinin, Feb 25 2015 */
(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
CROSSREFS
Cf. A066165. Apart from initial terms and signs, same as A007113.
Cf. A343579.
Sequence in context: A292122 A264417 A348311 * A007113 A052804 A267652
KEYWORD
nonn,nice,easy
AUTHOR
Len Smiley, Dec 12 2001
STATUS
approved