A186760 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)

0, 1, 2, 7, 33, 188, 1247, 9448, 80623, 765926, 8022139, 91872328, 1142384735, 15330003154, 220847064955, 3399884265524, 55705822616383, 967921774366510, 17778279366693179, 344189681672898400, 7005438733866799999, 149547115419379439978, 3341127481398057119515

Offset: 0

Author

Emeric Deutsch, Feb 26 2011

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

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

Programs

Maple

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);

Mathematica

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

Formula

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

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

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