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A130915 Number of permutations in the symmetric group S_n in which cycle lengths are odd and greater than 1.

%I A130915 #31 Jan 18 2024 12:06:40
%S A130915 1,0,0,2,0,24,40,720,2688,42560,245376,4072320,31672960,569935872,
%T A130915 5576263680,109492807424,1290701905920,27616577064960,380852962029568,
%U A130915 8845627365089280,139696582370328576,3506062524305162240,62387728088875499520,1684340707284076756992
%N A130915 Number of permutations in the symmetric group S_n in which cycle lengths are odd and greater than 1.
%H A130915 Alois P. Heinz, <a href="/A130915/b130915.txt">Table of n, a(n) for n = 0..450</a> (terms n = 1..200 from Vincenzo Librandi)
%F A130915 E.g.f.: exp(-x)*sqrt((1+x)/(1-x)).
%F A130915 a(n) ~ 2*n^n/exp(n+1). - _Vaclav Kotesovec_, Oct 08 2013
%F A130915 a(n) = (-1)^n*Sum_{k = 0..n} (1 if n = k, otherwise (-1)^(n + k)*(n - k)!*Sum_{i = 1..n - k} Sum_{j = i..n - k} 2^(j - i)*Stirling1(j, i)*binomial(n - k - 1, j - 1)/j!*binomial(n, k)). - _Detlef Meya_, Jan 18 2024
%F A130915 a(n) = (n-1)*(n-2)*(a(n-2)+a(n-3)) for n>=3. - _Alois P. Heinz_, Jan 18 2024
%e A130915 a(3)=2 because we have (123) and (132).
%p A130915 g:=exp(-x)*sqrt((1+x)/(1-x)): gser:=series(g,x=0,30): seq(factorial(n)*coeff(gser,x,n),n=0..20); # _Emeric Deutsch_, Aug 25 2007
%p A130915 # second Maple program:
%p A130915 a:= proc(n) option remember;
%p A130915       `if`(n<3, 1/2, a(n-2)+a(n-3))*(n-1)*(n-2)
%p A130915     end:
%p A130915 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jan 18 2024
%t A130915 nn=20;Drop[Range[0,nn]!CoefficientList[Series[((1+x)/(1-x))^(1/2)Exp[-x],{x,0,nn}],x],1]  (* _Geoffrey Critzer_, Dec 15 2012 *)
%t A130915 a[n_] := (-1)^n*Sum[If[n==k, 1, (-1)^(n + k)*(n - k)!*Sum[Sum[2^(j - i)*StirlingS1[j, i]*Binomial[n - k - 1, j - 1]/j!, {j, i, n - k}], {i, 1, n - k}]*Binomial[n, k]], {k, 0, n}]; Flatten[Table[a[n], {n, 1, 20}]] (* _Detlef Meya_, Jan 18 2024 *)
%o A130915 (PARI) my(x='x+O('x^33)); Vec(serlaplace(exp(-x)*sqrt((1+x)/(1-x)))) \\ _Joerg Arndt_, Jan 18 2024
%Y A130915 Cf. A000246, A000166.
%K A130915 easy,nonn
%O A130915 0,4
%A A130915 _Vladeta Jovovic_, Aug 23 2007
%E A130915 More terms from _Emeric Deutsch_, Aug 25 2007
%E A130915 a(0)=1 prepended by _Alois P. Heinz_, Jan 18 2024