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A349426
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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).
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1
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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
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OFFSET
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3,1
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COMMENTS
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A nontrivial round means the same as a ring or circle consisting of more than one child.
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)
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LINKS
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FORMULA
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E.g.f.: (1 - x)^(-x*t) * exp(-x^2*t).
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EXAMPLE
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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.
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MATHEMATICA
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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]}]
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CROSSREFS
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Row sums give A066165 (variant of Stanley's children's game).
Right border element of row n is A166334(n/3) for each n divisible by 3.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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