A066165 -id:A066165 - 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.

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

Offset: 2

Len Smiley, Dec 12 2001

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

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

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

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.

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

Cf. A343579.

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

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 *)

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 */

a(n)=if(n<0,0,n!*polcoeff(-1+1/(1-x+x*O(x^n))^x,n))

{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

a(n) = n!*sum(k=0, n\2, abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, May 10 2022

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

A349426 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k nontrivial rounds; n >= 3, 1 <= k <= floor(n/3).

Original entry on oeis.org

3, 8, 30, 144, 90, 840, 840, 5760, 7280, 45360, 66528, 7560, 403200, 657720, 151200, 3991680, 7064640, 2356200, 43545600, 82285632, 34890240, 1247400, 518918400, 1035365760, 521080560, 43243200, 6706022400, 14013679680, 8034586560, 1059458400

Offset: 3

Steven Finch, Nov 17 2021

A nontrivial round means the same as a ring or circle consisting of more than one child.

			Triangle starts:
[3]           3;
[4]           8;
[5]          30;
[6]         144,          90;
[7]         840,         840;
[8]        5760,        7280;
[9]       45360,       66528,       7560;
[10]     403200,      657720,     151200;
[11]    3991680,     7064640,    2356200;
[12]   43545600,    82285632,   34890240,    1247400;
[13]  518918400,  1035365760,  521080560,   43243200;
[14] 6706022400, 14013679680, 8034586560, 1059458400;
...
For n = 6, there are 144 ways to make one round and 90 ways to make two rounds.

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

Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Row sums give A066165 (variant of Stanley's children's game).
Column 1 gives A001048.
Right border element of row n is A166334(n/3) for each n divisible by 3.
Cf. A066166, A349280 (correspond to Stanley's original game).

f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t) Exp[-x^2 t], {x, 0, n}, {t, 0, k}]
Table[f[k, n], {n, 2, 14}, {k, 1, Floor[n/3]}]

E.g.f.: (1 - x)^(-x*t) * exp(-x^2*t).

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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.

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A349426 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k nontrivial rounds; n >= 3, 1 <= k <= floor(n/3).

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