A226170 The number of cycles over all even permutations of {1,2,...,n}.

0, 1, 2, 5, 26, 134, 894, 6474, 55152, 510768, 5334480, 60090480, 745035840, 9881421120, 141972324480, 2166467990400, 35410730342400, 611048958105600, 11198955424204800, 215604729694771200, 4379675205233664000, 93061582839880704000, 2075454840671815680000

Offset

0,3

Comments

a(n) = Sum Stirling1(n,k)*k where the sum is taken over even k for even n and odd k for odd n.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

a(n) = Sum_{k=1..n, 2|(n+k)} k*Stirling1(n,k).

E.g.f.: ((1+x)*log(1+x)-log(1-x)/(1-x))/2. - Vaclav Kotesovec, May 30 2013

a(n) ~ n! * (log(n) + gamma)/2, where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, May 30 2013

Example

a(3) = 5 because the even permutations of [3]: (1)(2)(3), (1,2,3), (1,3,2) have a total of 5 cycles.

Maple

with(combinat):
a:= n-> add(`if`(irem(n+k, 2)=0, k*stirling1(n, k), 0), k=1..n):
seq(a(n), n=0..25);  # Alois P. Heinz, May 29 2013

Mathematica

nn = 20; Range[0, nn]! CoefficientList[Series[D[Cosh[y Log[(1 - x^2)^(-1/2)]] Exp[y Log[((1 + x)/(1 - x))^(1/2)]], y] /. y -> 1, {x, 0, nn}], x]
With[{nmax = 30}, CoefficientList[Series[((1+x)*Log[1+x]-Log[1-x]/(1-x))/2, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Sep 04 2018 *)

Prog

(PARI) x='x+O('x^30); concat([0], Vec(serlaplace( ((1+x)*log(1+x)-log(1-x)/(1-x))/2 ))) \\ G. C. Greubel, Sep 04 2018

Crossrefs

Cf. A048994.

Sequence in context: A214951 A333004 A120762 * A371614 A334639 A072268

Adjacent sequences: A226167 A226168 A226169 * A226171 A226172 A226173

Keyword

nonn

Author

Geoffrey Critzer, May 29 2013

Status

approved