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A162972 Number of cycles in all non-derangement permutations of {1,2,...,n}.

%I A162972 #15 Jul 26 2022 13:43:19
%S A162972 1,2,9,38,210,1339,9870,82368,768432,7926903,89610070,1101767732,
%T A162972 14639237184,209048293375,3192959638778,51943905125760,
%U A162972 896723236236864,16373101528868943,315259605244694574,6384318171252621716,135651088007338895680,3017472066675257000775
%N A162972 Number of cycles in all non-derangement permutations of {1,2,...,n}.
%C A162972 a(n) = Sum(k*A162971(n,k), k=1..n).
%H A162972 Vincenzo Librandi, <a href="/A162972/b162972.txt">Table of n, a(n) for n = 1..200</a>
%F A162972 E.g.f.: (z*exp(-z) + (exp(-z)-1)*log(1-z)) / (1-z).
%F A162972 a(n) ~ n! * ((1-exp(-1))*(log(n) + gamma) + exp(-1)), where gamma is the Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 02 2013
%F A162972 Conjecture D-finite with recurrence a(n) +(-3*n+8)*a(n-1) +(n-3)*(3*n-14)*a(n-2) +(-n^3+19*n^2-98*n+156)*a(n-3) +(-4*n^3+56*n^2-258*n+395)*a(n-4) +(-6*n^3+84*n^2-395*n+624)*a(n-5) -(n-5)*(4*n^2-41*n+106)*a(n-6) -(n-5)*(n-6)^2*a(n-7)=0. - _R. J. Mathar_, Jul 26 2022
%e A162972 a(3) = 9 because in the 4 non-derangement permutations of {1,2,3,4}, namely (1)(2)(3), (1)(23), (12)(3), (13)(2), we have a total of 3 + 2 + 2 + 2 = 9 cycles.
%p A162972 g := (z*exp(-z)+(exp(-z)-1)*ln(1-z))/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 1 .. 22);
%t A162972 Rest[CoefficientList[Series[(x*Exp[-x]+(Exp[-x]-1)*Log[1-x])/(1-x), {x, 0, 20}], x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Oct 02 2013 *)
%Y A162972 Cf. A162971.
%K A162972 nonn
%O A162972 1,2
%A A162972 _Emeric Deutsch_, Jul 22 2009
%E A162972 a(21)-a(22) from _Vincenzo Librandi_, Oct 04 2013