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
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
Programs
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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;
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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)