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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
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
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.
Sequence in context: A025080 A024468 A247355 * A134683 A067853 A086590
KEYWORD
nonn,tabf
AUTHOR
Steven Finch, Nov 13 2021
STATUS
approved