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A326241 Number of degree-n even permutations of order dividing 12 2
1, 1, 1, 3, 12, 36, 216, 1296, 10368, 78912, 634896, 5572656, 51817536, 477672768, 8268884352, 101752505856, 1417554660096, 20985416983296, 344834432195328, 5096129755468032, 70148917686998016 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

REFERENCES

J. Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, Inc. New York, 1958 (Chap 4, Problem 22).

LINKS

Robert Israel, Table of n, a(n) for n = 0..482

Lev Glebsky, Melany Licón, Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.

FORMULA

E.g.f.: (1/2)*exp(x + (1/2)*x^2 + (1/3)*x^3 + (1/4)*x^4 + (1/6)*x^6+(1/12)*x^(12)) + (1/2)*exp(x - (1/2)*x^2 + (1/3)*x^3 - (1/4)*x^4 - (1/6)*x^6-(1/12)*x^(12)).

EXAMPLE

For n=3 the a(3)=3 solutions are (1), (1, 2, 3), (1, 3, 2) (permutations in cyclic notation).

MAPLE

E:= (1/2)*exp(x + (1/2)*x^2 + (1/3)*x^3 + (1/4)*x^4 + (1/6)*x^6+(1/12)*x^(12)) + (1/2)*exp(x - (1/2)*x^2 + (1/3)*x^3 - (1/4)*x^4 - (1/6)*x^6-(1/12)*x^(12)):

S:= series(E, x, 31):

seq(coeff(S, x, i)*i!, i=0..30); # Robert Israel, Jul 08 2019

MATHEMATICA

With[{nn = 22}, CoefficientList[Series[1/2 Exp[x + x^2/2 + x^3/3 + x^4/4 + x^6/6 +x^12/12]+1/2 Exp[x - x^2/2 + x^3/3 - x^4/4 - x^6/6 - x^12/12], {x, 0, nn}], x]*Range[0, nn]!]

CROSSREFS

Cf. A053502, A326242, A000704, A061130, A061131, A061132, A048099, A051695, A061133, A061134, A061135, A326242.

Sequence in context: A004661 A074430 A064028 * A110950 A102744 A145951

Adjacent sequences:  A326238 A326239 A326240 * A326242 A326243 A326244

KEYWORD

easy,nonn

AUTHOR

Luis Manuel Rivera Martínez, Jul 06 2019

STATUS

approved

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Last modified September 28 07:55 EDT 2020. Contains 337394 sequences. (Running on oeis4.)