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A178575
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Number of permutations of {1,2,...,3n} whose cycle lengths are all divisible by 3.
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5
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1, 2, 160, 62720, 68992000, 163235072000, 710399033344000, 5129081020743680000, 57096929922918645760000, 927825111247427993600000000, 21095031729321522862489600000000, 648714415740095471067280179200000000, 26246985260844262759382156050432000000000
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OFFSET
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0,2
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REFERENCES
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Herbert S. Wilf, Generatingfunctiontology, page 209
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LINKS
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FORMULA
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a(n) = (-1)^(n/3)*binomial(-1/3,n/3)*n!.
E.g.f.: 1/(1-x^3)^(1/3).
a(n) = ((3*n)!/(n!*3^n))*Product_{i=1..n-1} (1+3*i) (from the Wilf reference).
D-finite with recurrence: a(n) = (3*n-1)*(3*n-2)^2*a(n-1), a(0)=1. - Georg Fischer, Jul 02 2021 (from the 3rd formula)
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EXAMPLE
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a(1) = 2 because we have (123) and (132).
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MAPLE
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a:= n-> factorial(3*n)*(mul(1+3*i, i = 1 .. n-1))/(factorial(n)*3^n): seq(a(n), n = 0 .. 11);
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MATHEMATICA
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Table[(-1)^(n/3) Binomial[-1/3, n/3]n!, {n, 0, 30, 3}]
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PROG
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(PARI) v=Vec(serlaplace(1/(1-x^3+O(x^50))^(1/3))); vector(#v\3, n, v[3*n-2])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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