A033030 Derangement numbers d(n,3) where d(n,k) = k(n-1)(d(n-1,k) + d(n-2,k)), with d(0,k) = 1 and d(1,k) = 0.

1, 0, 3, 18, 189, 2484, 40095, 766422, 16936857, 424878696, 11929019931, 370616958810, 12624017298453, 467806833261468, 18736803171836919, 806593620214132254, 37139869052368612785, 1821430208283971761872, 94787073944153359107507, 5216859224231615866946466

Offset: 0

Author

N. J. A. Sloane, Nov 02 2003

Keywords

• nonn
• easy

Examples

			3= 3*(1+0), 18 =6*(0+3), 189=9*(18+3), 2484=12*(189+18)... [From _Gary Detlefs_, May 16 2010]
		

Links

• Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013

Crossrefs

d(n, 1) gives A000166, d(n, 2) gives A053871, d(n, 4) gives A088991, d(n, 5) gives A088992.

Cf. A381484.

Programs

Maple

k := 3; d := proc(n) global k; option remember; if n = 0 then RETURN(1) end if; if n = 1 then RETURN(0) end if; k*(n - 1)*(d(n - 1) + d(n - 2)) end proc;

Mathematica

d[n_, k_] := d[n, k] = k(n-1)(d[n-1, k] + d[n-2, k]);
d[0, ] = 1; d[1, ] = 0;
a[n_] := d[n, 3];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 20 2023 *)

Formula

Inverse binomial transform of A007559. E.g.f.: exp(-x)/(1-3*x)^(1/3). - Vladeta Jovovic, Dec 17 2003

a(n) = 3(n-1)(a(n-1)+a(n-2)), n>1. - Gary Detlefs, May 16 2010

a(n) ~ Gamma(2/3) * 3^(n + 1/2) * n^(n-1/6) / (sqrt(2*Pi) * exp(n + 1/3)). - Vaclav Kotesovec, Oct 31 2017

From Seiichi Manyama, Apr 23 2025: (Start)

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A381484.

a(n) = (-1)^n * n! * Sum_{k=0..n} 3^k * binomial(-1/3,k)/(n-k)!. (End)