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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).
1
3, 8, 30, 144, 90, 840, 840, 5760, 7280, 45360, 66528, 7560, 403200, 657720, 151200, 3991680, 7064640, 2356200, 43545600, 82285632, 34890240, 1247400, 518918400, 1035365760, 521080560, 43243200, 6706022400, 14013679680, 8034586560, 1059458400
OFFSET
3,1
COMMENTS
A nontrivial round means the same as a ring or circle consisting of more than one child.
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
E.g.f.: (1 - x)^(-x*t) * exp(-x^2*t).
EXAMPLE
Triangle starts:
[3] 3;
[4] 8;
[5] 30;
[6] 144, 90;
[7] 840, 840;
[8] 5760, 7280;
[9] 45360, 66528, 7560;
[10] 403200, 657720, 151200;
[11] 3991680, 7064640, 2356200;
[12] 43545600, 82285632, 34890240, 1247400;
[13] 518918400, 1035365760, 521080560, 43243200;
[14] 6706022400, 14013679680, 8034586560, 1059458400;
...
For n = 6, there are 144 ways to make one round and 90 ways to make two rounds.
MATHEMATICA
f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t) Exp[-x^2 t], {x, 0, n}, {t, 0, k}]
Table[f[k, n], {n, 2, 14}, {k, 1, Floor[n/3]}]
CROSSREFS
Row sums give A066165 (variant of Stanley's children's game).
Column 1 gives A001048.
Right border element of row n is A166334(n/3) for each n divisible by 3.
Cf. A066166, A349280 (correspond to Stanley's original game).
Sequence in context: A074501 A009123 A066764 * A059171 A261766 A078619
KEYWORD
nonn,tabf
AUTHOR
Steven Finch, Nov 17 2021
STATUS
approved