%I #46 May 03 2023 17:05:42

%S 2,3,8,12,30,60,144,330,120,840,2100,1260,5760,15344,11760,1680,45360,

%T 127008,113400,30240,403200,1176120,1169280,428400,30240,3991680,

%U 12054240,13000680,5821200,831600,43545600,135508032,155923680,80415720,16632000,665280

%N Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k rounds; n >= 2, 1 <= k <= floor(n/2).

%C A round means the same as a directed ring or circle.

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

%H Steven Finch, <a href="https://arxiv.org/abs/2111.14487">Rounds, Color, Parity, Squares</a>, arXiv:2111.14487 [math.CO], 2021.

%F G.f.: (1 - x)^(-x*t).

%F T(n, k) = binomial(n, k)*k!*|Stirling1(n-k, k)|. - _Mélika Tebni_, May 03 2023

%F The above formula can also be written as T(n, k) = A008279(n, k)*A331327(n, k) or as T(n, k) = A265609(n + 1, k)*A331327(n, k). - _Peter Luschny_, May 03 2023

%e Triangle starts:

%e [2] 2;

%e [3] 3;

%e [4] 8, 12;

%e [5] 30, 60;

%e [6] 144, 330, 120;

%e [7] 840, 2100, 1260;

%e [8] 5760, 15344, 11760, 1680;

%e [9] 45360, 127008, 113400, 30240;

%e ...

%e For n = 4, there are 8 ways to make one round and 12 ways to make two rounds.

%p ser := series((1 - x)^(-x*t), x, 20): xcoeff := n -> coeff(ser, x, n):

%p T := (n, k) -> n!*coeff(xcoeff(n), t, k):

%p seq(seq(T(n, k), k = 1..iquo(n,2)), n = 2..12); # _Peter Luschny_, Nov 13 2021

%p # second Maple program:

%p A349280 := (n,k) -> binomial(n,k)*k!*abs(Stirling1(n-k,k)):

%p seq(print(seq(A349280(n,k), k=1..iquo(n,2))), n=2..12); # _Mélika Tebni_, May 03 2023

%t f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t), {x, 0, n}, {t, 0, k}]

%t Table[f[k, n], {n, 2, 12}, {k, 1, Floor[n/2]}]

%Y Row sums give A066166 (Stanley's children's game).

%Y Column 1 gives A001048.

%Y Right border element of row n is A001813(n/2) = |A067994(n)| for even n.

%Y Cf. A008279, A265609, A331327.

%K nonn,tabf

%O 2,1

%A _Steven Finch_, Nov 13 2021