A130619 - OEIS

%I #53 Dec 11 2022 02:14:03

%S 0,1,2,6,25,127,768,5401,43335,390783,3913231,43088876,517457295,

%T 6730858066,94275101800,1414643984295,22641034606786,384991863417162,

%U 6931268185493211,131716736558977795,2634719723042973062

%N Let M(n) = {{n, 0, 1}, {1, 0, 0}, {0, 1, 0}}, then a(n) is the upper-right term of M(n)*M(n-1)*...*M(1).

%C Equivalently, a(n) is the middle-left term of M(1)*M(2)*...*M(n). - _Jianing Song_, Sep 24 2018

%F a(n) = n*a(n-1) + a(n-3) for n >= 3. - _David A. Corneth_, Sep 24 2018

%F E.g.f.: A'''(x), where A'''(x) = A(x)/(1 - x) + x + x^2, A(0) = A'(0) = A''(0) = 0.

%t M[n_] := {{n, 0, 1}, {1, 0, 0}, {0, 1, 0}};

%t v[0] = {0, 0, 1};

%t v[n_] := v[n] = M[n].v[n - 1];

%t a = Table[v[n][[1]], {n, 0, 30}]

%o (PARI) a(n) = if(n<3, n, n*a(n-1) + a(n-3)) \\ _Jianing Song_, Sep 24 2018

%o (PARI) M(n) = [n, 0, 1; 1, 0, 0; 0, 1, 0];

%o lista(nn) = {v = [0, 0, 1]; for (n=0, nn, Mn = M(n); v = vector(3, k, sum(i=1, 3, Mn[i,k]*v[i])); print1(v[1], ", "););} \\ _Michel Marcus_, Sep 24 2018

%o (PARI) a=vector(50); a[1]=1; a[2]=2; a[3]=6; for(n=4, #a, a[n]=n*a[n-1]+a[n-3]); concat(0, a) \\ _Altug Alkan_, Sep 24 2018

%Y Cf. A001053.

%K nonn,easy

%O 0,3

%A _Roger L. Bagula_, Jun 18 2007

%E Edited, new name, and offset corrected by _Jianing Song_, Sep 24 2018