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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). 2
 2, 3, 8, 12, 30, 60, 144, 330, 120, 840, 2100, 1260, 5760, 15344, 11760, 1680, 45360, 127008, 113400, 30240, 403200, 1176120, 1169280, 428400, 30240, 3991680, 12054240, 13000680, 5821200, 831600, 43545600, 135508032, 155923680, 80415720, 16632000, 665280 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS A round means the same as a directed ring or circle. REFERENCES R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2) LINKS Table of n, a(n) for n=2..37. Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021. FORMULA G.f.: (1 - x)^(-x*t). T(n, k) = binomial(n, k)*k!*|Stirling1(n-k, k)|. - Mélika Tebni, May 03 2023 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 EXAMPLE Triangle starts: [2] 2; [3] 3; [4] 8, 12; [5] 30, 60; [6] 144, 330, 120; [7] 840, 2100, 1260; [8] 5760, 15344, 11760, 1680; [9] 45360, 127008, 113400, 30240; ... For n = 4, there are 8 ways to make one round and 12 ways to make two rounds. MAPLE ser := series((1 - x)^(-x*t), x, 20): xcoeff := n -> coeff(ser, x, n): T := (n, k) -> n!*coeff(xcoeff(n), t, k): seq(seq(T(n, k), k = 1..iquo(n, 2)), n = 2..12); # Peter Luschny, Nov 13 2021 # second Maple program: A349280 := (n, k) -> binomial(n, k)*k!*abs(Stirling1(n-k, k)): seq(print(seq(A349280(n, k), k=1..iquo(n, 2))), n=2..12); # Mélika Tebni, May 03 2023 MATHEMATICA f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t), {x, 0, n}, {t, 0, k}] Table[f[k, n], {n, 2, 12}, {k, 1, Floor[n/2]}] CROSSREFS Row sums give A066166 (Stanley's children's game). Column 1 gives A001048. Right border element of row n is A001813(n/2) = |A067994(n)| for even n. Cf. A008279, A265609, A331327. Sequence in context: A025080 A024468 A247355 * A134683 A067853 A086590 Adjacent sequences: A349277 A349278 A349279 * A349281 A349282 A349283 KEYWORD nonn,tabf AUTHOR Steven Finch, Nov 13 2021 STATUS approved

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Last modified September 10 09:32 EDT 2024. Contains 375786 sequences. (Running on oeis4.)