cp's OEIS Frontend

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.

%I A066166 #49 May 10 2022 11:45:18
%S A066166 2,3,20,90,594,4200,34544,316008,3207240,35699400,432690312,
%T A066166 5672581200,79991160144,1207367605080,19423062612480,331770360922560,
%U A066166 5997105160795584,114373526841360000,2295170834453089920
%N 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.
%C A066166 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
%D A066166 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)
%H A066166 Harvey P. Dale, <a href="/A066166/b066166.txt">Table of n, a(n) for n = 2..250</a>
%H A066166 Steven Finch, <a href="https://arxiv.org/abs/2111.14487">Rounds, Color, Parity, Squares</a>, arXiv:2111.14487 [math.CO], 2021.
%H A066166 P. Flajolet, S. Gerhold and B. Salvy, <a href="http://arXiv.org/abs/math/0501379">On the non-holonomic character of logarithms, powers and the n-th prime function</a>, arXiv:math/0501379 [math.CO], 2005.
%F A066166 E.g.f.: -1+1/(1-x)^x.
%F A066166 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
%F A066166 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
%F A066166 E.g.f.: Sum_{n>=1} x^n/n! * Product_{k=0..n-1} (k + x). - _Paul D. Hanna_, Oct 26 2015
%F A066166 a(n) = n! * Sum_{k=0..floor(n/2)} |Stirling1(n-k,k)|/(n-k)!. - _Seiichi Manyama_, May 10 2022
%e A066166 a(4)=20: 12 ways to make 2 hugs, 8 ways to make a 3-ring.
%t A066166 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 *)
%o A066166 (PARI) a(n)=if(n<0,0,n!*polcoeff(-1+1/(1-x+x*O(x^n))^x,n))
%o A066166 (PARI) {a(n) = n!*polcoeff( sum(m=1,n, x^m/m! * prod(k=0,m-1,x + k) +x*O(x^n) ), n)}
%o A066166 for(n=2,20, print1(a(n),", ")) \\ _Paul D. Hanna_, Oct 26 2015
%o A066166 (PARI) a(n) = n!*sum(k=0, n\2, abs(stirling(n-k, k, 1))/(n-k)!); \\ _Seiichi Manyama_, May 10 2022
%o A066166 (Maxima)
%o A066166 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);
%o A066166 makelist(a(n),n,2,10); /* _Vladimir Kruchinin_, Feb 25 2015 */
%o A066166 (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
%Y A066166 Cf. A066165. Apart from initial terms and signs, same as A007113.
%Y A066166 Cf. A343579.
%K A066166 nonn,nice,easy
%O A066166 2,1
%A A066166 _Len Smiley_, Dec 12 2001