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.

2, 3, 20, 90, 594, 4200, 34544, 316008, 3207240, 35699400, 432690312, 5672581200, 79991160144, 1207367605080, 19423062612480, 331770360922560, 5997105160795584, 114373526841360000, 2295170834453089920

Offset: 2

Author

Len Smiley, Dec 12 2001

Keywords

• nonn
• nice
• easy

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

Examples

			a(4)=20: 12 ways to make 2 hugs, 8 ways to make a 3-ring.
		

References

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

Links

• Harvey P. Dale, Table of n, a(n) for n = 2..250
• 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.

Crossrefs

Cf. A066165. Apart from initial terms and signs, same as A007113.

Cf. A343579.

Programs

Magma

m:=30; R:=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

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 *)

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 */

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

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