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A053498
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Number of degree-n permutations of order dividing 8.
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4
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1, 1, 2, 4, 16, 56, 256, 1072, 11264, 78976, 672256, 4653056, 49810432, 433429504, 4448608256, 39221579776, 607251736576, 7244686764032, 101611422797824, 1170362064019456, 19281174853615616, 261583327556386816, 4084459360167657472, 54366023748591386624
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OFFSET
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0,3
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
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LINKS
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FORMULA
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E.g.f.: exp(x + x^2/2 + x^4/4 + x^8/8).
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MAPLE
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a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 2, 4, 8])))
end:
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MATHEMATICA
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CoefficientList[Series[Exp[x+x^2/2+x^4/4+x^8/8], {x, 0, 23}], x]*Range[0, 23]! (* Jean-François Alcover, Mar 24 2014 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x +x^2/2 +x^4/4 +x^8/8) )) \\ G. C. Greubel, May 14 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +x^2/2 +x^4/4 +x^8/8) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 14 2019
(Sage) m = 30; T = taylor(exp(x +x^2/2 +x^4/4 +x^8/8), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 14 2019
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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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