A082425 - OEIS

%I #33 Apr 27 2025 11:50:52

%S 1,1,5,27,169,1217,9939,90871,920069,10222989,123698167,1619321459,

%T 22805443881,343835923129,5525934478859,94309281772527,

%U 1703461402016269,32465970250192421,651123070017747999,13707854105636799979,302258183029291439537,6966331456484621749329

%N a(1)=1, a(n) = -1 + n*Sum_{j=1..n-1} a(j).

%H G. C. Greubel, <a href="/A082425/b082425.txt">Table of n, a(n) for n = 1..445</a>

%F For n >= 2, a(n) = floor(n*(3-e)*n!).

%F a(n) = n*A056543(n) - 1, n > 1. - _Vladeta Jovovic_, Apr 26 2003

%F From _Peter Bala_, Jul 09 2008: (Start)

%F In the following remarks we use an offset of 1, i.e., a(1) = 1, a(2) = 1, a(3) = 5, ... .

%F   For n >= 2, a(n) = n*n!*Sum_{k = 2..n} 1/(k*(k-1)*k!).

%F   For n >= 2, a(n) = 3*n*n! - Sum_{k = 0..n} (k+1)!*binomial(n,k).

%F   Limit_{n -> oo} a(n)/(n*n!) = 3 - e.

%F   E.g.f.: 1 + t + (3*t - exp(t))/(1-t)^2.

%F   a(n) = A083746(n+2) - A001339(n).

%F Recurrence relation: a(1) = 1, a(2) = 1, a(3) = 5, a(n) = (n+2)*a(n-1) - (n-1)*a(n-2) for n >= 4.

%F Recurrence relation: a(1) = 1, a(2) = 1, a(n) = (n^2*a(n-1) + 1)/(n-1) for n >= 3.

%F The recurrence relation x(n) = (n^2*x(n-1) - 1)/(n-1), for n >= 2, has the general solution x(n) = n*n!*x(1) - a(n); particular solutions are A007808 (x(1) = 1) and A001339 (x(1) = 3). (End)

%p a:= n -> n*n!*add(1/(k*(k-1)*k!), k = 2..n): seq(a(n), n = 2..20); # _Peter Bala_, Jul 09 2008

%t a[n_]:= a[n]= If[n<3, 1, -1 +n*Sum[a[j], {j,n-1}]];

%t Table[a[n], {n,40}] (* _G. C. Greubel_, Feb 03 2024 *)

%o (Magma) [n le 2 select 1 else (n^2*Self(n-1) +1)/(n-1): n in [1..30]]; // _G. C. Greubel_, Feb 03 2024

%o (SageMath)

%o @CachedFunction # a = A082425

%o def a(n): return 1 if (n==1) else -1 + n*sum(a(j) for j in range(1,n))

%o [a(n) for n in range(1,41)] # _G. C. Greubel_, Feb 03 2024

%Y Cf. A001339, A056543, A083746.

%Y Cf. A007808, A074143, A082427, A082428, A082430, A383436, A383437.

%K nonn

%O 1,3

%A _Benoit Cloitre_, Apr 24 2003

%E Offset corrected by _G. C. Greubel_, Feb 03 2024