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

1, 2, 9, 38, 210, 1339, 9870, 82368, 768432, 7926903, 89610070, 1101767732, 14639237184, 209048293375, 3192959638778, 51943905125760, 896723236236864, 16373101528868943, 315259605244694574, 6384318171252621716, 135651088007338895680, 3017472066675257000775

Offset

1,2

Comments

a(n) = Sum(k*A162971(n,k), k=1..n).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Formula

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

a(n) ~ n! * ((1-exp(-1))*(log(n) + gamma) + exp(-1)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 02 2013

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

Example

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.

Maple

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);

Mathematica

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 *)

Crossrefs

Cf. A162971.

Sequence in context: A057647 A249925 A370397 * A202832 A069724 A132961

Adjacent sequences: A162969 A162970 A162971 * A162973 A162974 A162975

Keyword

nonn

Author

Emeric Deutsch, Jul 22 2009

Extensions

a(21)-a(22) from Vincenzo Librandi, Oct 04 2013

Status

approved