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A053503 Number of degree-n permutations of order dividing 16. 3
1, 1, 2, 4, 16, 56, 256, 1072, 11264, 78976, 672256, 4653056, 49810432, 433429504, 4448608256, 39221579776, 1914926104576, 29475151020032, 501759779405824, 6238907914387456, 120652091860975616, 1751735807564578816, 29062253310781161472, 398033706586943258624 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Differs from A005388 first at n=32. - Alois P. Heinz, Feb 14 2013
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
LINKS
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
FORMULA
E.g.f.: exp(x + x^2/2 + x^4/4 + x^8/8 + x^16/16).
MAPLE
a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
add(mul(n-i, i=1..2^j-1)*a(n-2^j), j=0..4)))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Feb 14 2013
MATHEMATICA
a[n_]:= a[n] =If[n<0, 0, If[n==0, 1, Sum[Product[n-i, {i, 1, 2^j-1}]* a[n-2^j], {j, 0, 4}]]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 19 2014, after Alois P. Heinz *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^4/4 +x^8/8 + x^16/16], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 15 2019 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x + x^2/2 + x^4/4 + x^8/8 + x^16/16) )) \\ G. C. Greubel, May 15 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 + x^4/4 + x^8/8 + x^16/16) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019
(Sage) m = 30; T = taylor(exp(x + x^2/2 + x^4/4 + x^8/8 + x^16/16), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019
CROSSREFS
Column k=16 of A008307.
Sequence in context: A001472 A053498 A005388 * A308381 A153957 A068789
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 15 2000
STATUS
approved

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