A211880 Number of permutations of n elements with no fixed points and largest cycle of length 3.

0, 0, 0, 2, 0, 20, 40, 210, 1120, 4760, 25200, 157850, 800800, 5345340, 35035000, 222472250, 1648046400, 12000388400, 88529240800, 720929459250, 5786188408000, 48072795270500, 424300329453000, 3731123025279650, 34083741984292000, 323768324084205000

Offset: 0

Author

Alois P. Heinz, Feb 13 2013

Keywords

• nonn

Comments

a(n) = A055814(n) - A123023(n). - Vaclav Kotesovec, Oct 09 2013

Examples

			a(3) = 2: (2,3,1), (3,1,2).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250

Crossrefs

Column k=3 of A211871.

Cf. A055814, A123023.

Programs

Maple

egf:= (exp(x^3/3)-1)*exp(x^2/2):
a:= n-> n! *coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..30);

Mathematica

A[n_, k_] := A[n, k] = If[n < 0, 0, If[n == 0, 1,
     Sum[Product[n - i, {i, 1, j - 1}] A[n - j, k], {j, 2, k}]]];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
a[n_] := T[n, 3];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 03 2021, after Alois P. Heinz in A211871 *)

Formula

E.g.f.: (exp(x^3/3)-1) * exp(x^2/2).

Recurrence: (n-3)*a(n) = (n-1)*(2*n-5)*a(n-2) + (n-3)*(n-2)*(n-1)*a(n-3) - (n-3)*(n-2)*(n-1)*a(n-4) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-5). - Vaclav Kotesovec, Oct 09 2013