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A082425 a(1)=1, a(n) = -1 + n*Sum_{j=1..n-1} a(j).

%I A082425 #33 Apr 27 2025 11:50:52
%S A082425 1,1,5,27,169,1217,9939,90871,920069,10222989,123698167,1619321459,
%T A082425 22805443881,343835923129,5525934478859,94309281772527,
%U A082425 1703461402016269,32465970250192421,651123070017747999,13707854105636799979,302258183029291439537,6966331456484621749329
%N A082425 a(1)=1, a(n) = -1 + n*Sum_{j=1..n-1} a(j).
%H A082425 G. C. Greubel, <a href="/A082425/b082425.txt">Table of n, a(n) for n = 1..445</a>
%F A082425 For n >= 2, a(n) = floor(n*(3-e)*n!).
%F A082425 a(n) = n*A056543(n) - 1, n > 1. - _Vladeta Jovovic_, Apr 26 2003
%F A082425 From _Peter Bala_, Jul 09 2008: (Start)
%F A082425 In the following remarks we use an offset of 1, i.e., a(1) = 1, a(2) = 1, a(3) = 5, ... .
%F A082425   For n >= 2, a(n) = n*n!*Sum_{k = 2..n} 1/(k*(k-1)*k!).
%F A082425   For n >= 2, a(n) = 3*n*n! - Sum_{k = 0..n} (k+1)!*binomial(n,k).
%F A082425   Limit_{n -> oo} a(n)/(n*n!) = 3 - e.
%F A082425   E.g.f.: 1 + t + (3*t - exp(t))/(1-t)^2.
%F A082425   a(n) = A083746(n+2) - A001339(n).
%F A082425 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 A082425 Recurrence relation: a(1) = 1, a(2) = 1, a(n) = (n^2*a(n-1) + 1)/(n-1) for n >= 3.
%F A082425 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 A082425 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 A082425 a[n_]:= a[n]= If[n<3, 1, -1 +n*Sum[a[j], {j,n-1}]];
%t A082425 Table[a[n], {n,40}] (* _G. C. Greubel_, Feb 03 2024 *)
%o A082425 (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 A082425 (SageMath)
%o A082425 @CachedFunction # a = A082425
%o A082425 def a(n): return 1 if (n==1) else -1 + n*sum(a(j) for j in range(1,n))
%o A082425 [a(n) for n in range(1,41)] # _G. C. Greubel_, Feb 03 2024
%Y A082425 Cf. A001339, A056543, A083746.
%Y A082425 Cf. A007808, A074143, A082427, A082428, A082430, A383436, A383437.
%K A082425 nonn
%O A082425 1,3
%A A082425 _Benoit Cloitre_, Apr 24 2003
%E A082425 Offset corrected by _G. C. Greubel_, Feb 03 2024