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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).

%I A349426 #20 Dec 10 2021 05:55:14
%S A349426 3,8,30,144,90,840,840,5760,7280,45360,66528,7560,403200,657720,
%T A349426 151200,3991680,7064640,2356200,43545600,82285632,34890240,1247400,
%U A349426 518918400,1035365760,521080560,43243200,6706022400,14013679680,8034586560,1059458400
%N 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).
%C A349426 A nontrivial round means the same as a ring or circle consisting of more than one child.
%D A349426 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)
%H A349426 Steven Finch, <a href="https://arxiv.org/abs/2111.14487">Rounds, Color, Parity, Squares</a>, arXiv:2111.14487 [math.CO], 2021.
%F A349426 E.g.f.: (1 - x)^(-x*t) * exp(-x^2*t).
%e A349426 Triangle starts:
%e A349426 [3]           3;
%e A349426 [4]           8;
%e A349426 [5]          30;
%e A349426 [6]         144,          90;
%e A349426 [7]         840,         840;
%e A349426 [8]        5760,        7280;
%e A349426 [9]       45360,       66528,       7560;
%e A349426 [10]     403200,      657720,     151200;
%e A349426 [11]    3991680,     7064640,    2356200;
%e A349426 [12]   43545600,    82285632,   34890240,    1247400;
%e A349426 [13]  518918400,  1035365760,  521080560,   43243200;
%e A349426 [14] 6706022400, 14013679680, 8034586560, 1059458400;
%e A349426 ...
%e A349426 For n = 6, there are 144 ways to make one round and 90 ways to make two rounds.
%t A349426 f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t) Exp[-x^2 t], {x, 0, n}, {t, 0, k}]
%t A349426 Table[f[k, n], {n, 2, 14}, {k, 1, Floor[n/3]}]
%Y A349426 Row sums give A066165 (variant of Stanley's children's game).
%Y A349426 Column 1 gives A001048.
%Y A349426 Right border element of row n is A166334(n/3) for each n divisible by 3.
%Y A349426 Cf. A066166, A349280 (correspond to Stanley's original game).
%K A349426 nonn,tabf
%O A349426 3,1
%A A349426 _Steven Finch_, Nov 17 2021