%I #17 Jul 26 2022 16:18:40

%S 0,1,2,7,33,188,1247,9448,80623,765926,8022139,91872328,1142384735,

%T 15330003154,220847064955,3399884265524,55705822616383,

%U 967921774366510,17778279366693179,344189681672898400,7005438733866799999,149547115419379439978,3341127481398057119515

%N Number of cycles that are either nonincreasing or of length 1 in all permutations of {1,2,...,n}. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)<b(2)<b(3)<... .

%C a(n) = Sum(A186759(n,k), k=0..n).

%H Alois P. Heinz, <a href="/A186760/b186760.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f.: (1+z-exp(z)-log(1-z))/(1-z).

%F a(n) ~ n! * (log(n) + gamma + 2 - exp(1)), where gamma is the Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 08 2013

%F D-finite with recurrence a(n) +(-2*n-1)*a(n-1) +(n^2+2*n-4)*a(n-2) +(-2*n^2+6*n-3)*a(n-3) +((n-3)^2)*a(n-4)=0. - _R. J. Mathar_, Jul 26 2022

%e a(3) = 7 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), and (132) we have a total of 3 + 1 + 1 + 1 + 0 + 1 = 7 cycles that are either of length 1 or nonincreasing.

%p g := (1+z-exp(z)-ln(1-z))/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22);

%t CoefficientList[Series[(1+x-E^x-Log[1-x])/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 08 2013 *)

%Y Cf. A186754, A186755, A186756, A186757, A186758, A186759.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Feb 26 2011