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A349280 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).

%I A349280 #46 May 03 2023 17:05:42
%S A349280 2,3,8,12,30,60,144,330,120,840,2100,1260,5760,15344,11760,1680,45360,
%T A349280 127008,113400,30240,403200,1176120,1169280,428400,30240,3991680,
%U A349280 12054240,13000680,5821200,831600,43545600,135508032,155923680,80415720,16632000,665280
%N A349280 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 A349280 A round means the same as a directed ring or circle.
%D A349280 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)
%H A349280 Steven Finch, <a href="https://arxiv.org/abs/2111.14487">Rounds, Color, Parity, Squares</a>, arXiv:2111.14487 [math.CO], 2021.
%F A349280 G.f.: (1 - x)^(-x*t).
%F A349280 T(n, k) = binomial(n, k)*k!*|Stirling1(n-k, k)|. - _Mélika Tebni_, May 03 2023
%F A349280 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 A349280 Triangle starts:
%e A349280 [2]     2;
%e A349280 [3]     3;
%e A349280 [4]     8,     12;
%e A349280 [5]    30,     60;
%e A349280 [6]   144,    330,    120;
%e A349280 [7]   840,   2100,   1260;
%e A349280 [8]  5760,  15344,  11760,  1680;
%e A349280 [9] 45360, 127008, 113400, 30240;
%e A349280 ...
%e A349280 For n = 4, there are 8 ways to make one round and 12 ways to make two rounds.
%p A349280 ser := series((1 - x)^(-x*t), x, 20): xcoeff := n -> coeff(ser, x, n):
%p A349280 T := (n, k) -> n!*coeff(xcoeff(n), t, k):
%p A349280 seq(seq(T(n, k), k = 1..iquo(n,2)), n = 2..12); # _Peter Luschny_, Nov 13 2021
%p A349280 # second Maple program:
%p A349280 A349280 := (n,k) -> binomial(n,k)*k!*abs(Stirling1(n-k,k)):
%p A349280 seq(print(seq(A349280(n,k), k=1..iquo(n,2))), n=2..12); # _Mélika Tebni_, May 03 2023
%t A349280 f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t), {x, 0, n}, {t, 0, k}]
%t A349280 Table[f[k, n], {n, 2, 12}, {k, 1, Floor[n/2]}]
%Y A349280 Row sums give A066166 (Stanley's children's game).
%Y A349280 Column 1 gives A001048.
%Y A349280 Right border element of row n is A001813(n/2) = |A067994(n)| for even n.
%Y A349280 Cf. A008279, A265609, A331327.
%K A349280 nonn,tabf
%O A349280 2,1
%A A349280 _Steven Finch_, Nov 13 2021