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%I #21 Sep 05 2018 05:52:52
%S 0,1,2,5,26,134,894,6474,55152,510768,5334480,60090480,745035840,
%T 9881421120,141972324480,2166467990400,35410730342400,611048958105600,
%U 11198955424204800,215604729694771200,4379675205233664000,93061582839880704000,2075454840671815680000
%N The number of cycles over all even permutations of {1,2,...,n}.
%C a(n) = Sum Stirling1(n,k)*k where the sum is taken over even k for even n and odd k for odd n.
%H Alois P. Heinz, <a href="/A226170/b226170.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k=1..n, 2|(n+k)} k*Stirling1(n,k).
%F E.g.f.: ((1+x)*log(1+x)-log(1-x)/(1-x))/2. - _Vaclav Kotesovec_, May 30 2013
%F a(n) ~ n! * (log(n) + gamma)/2, where gamma is Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, May 30 2013
%e a(3) = 5 because the even permutations of [3]: (1)(2)(3), (1,2,3), (1,3,2) have a total of 5 cycles.
%p with(combinat):
%p a:= n-> add(`if`(irem(n+k, 2)=0, k*stirling1(n, k), 0), k=1..n):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, May 29 2013
%t 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]
%t 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 *)
%o (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
%Y Cf. A048994.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, May 29 2013
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