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

1, 1, 5, 27, 169, 1217, 9939, 90871, 920069, 10222989, 123698167, 1619321459, 22805443881, 343835923129, 5525934478859, 94309281772527, 1703461402016269, 32465970250192421, 651123070017747999, 13707854105636799979

Offset

1,3

Links

G. C. Greubel, Table of n, a(n) for n = 1..445

Formula

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

a(n) = n*A056543(n) - 1, n > 1. - Vladeta Jovovic, Apr 26 2003

From Peter Bala, Jul 09 2008: (Start)

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

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

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

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

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

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

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.

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

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)

Maple

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

Mathematica

a[n_]:= a[n]= If[n<3, 1, -1 +n*Sum[a[j], {j, n-1}]];
Table[a[n], {n, 40}] (* G. C. Greubel, Feb 03 2024 *)

Prog

(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
(SageMath)
@CachedFunction # a = A082425
def a(n): return 1 if (n==1) else -1 + n*sum(a(j) for j in range(1, n))
[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 03 2024

Crossrefs

Cf. A001339, A007808, A056543, A074143, A083746.

Sequence in context: A138772 A258789 A367259 * A202248 A225344 A109963

Adjacent sequences: A082422 A082423 A082424 * A082426 A082427 A082428

Keyword

nonn

Author

Benoit Cloitre, Apr 24 2003

Extensions

Offset corrected by G. C. Greubel, Feb 03 2024

Status

approved