A326241 Number of degree-n even permutations of order dividing 12

1, 1, 1, 3, 12, 36, 216, 1296, 10368, 78912, 634896, 5572656, 51817536, 477672768, 8268884352, 101752505856, 1417554660096, 20985416983296, 344834432195328, 5096129755468032, 70148917686998016

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 A372880 A102744

Adjacent sequences: A326238 A326239 A326240 * A326242 A326243 A326244

Keyword

easy,nonn

Author

Luis Manuel Rivera Martínez, Jul 06 2019

Status

approved