A007526 -id:A007526 - cp's OEIS frontend

A117643 a(n) = n*(a(n-1)-1) starting with a(0)=3.

3, 2, 2, 3, 8, 35, 204, 1421, 11360, 102231, 1022300, 11245289, 134943456, 1754264915, 24559708796, 368395631925, 5894330110784, 100203611883311, 1803665013899580, 34269635264092001, 685392705281840000

Offset: 0

Henry Bottomley, Apr 10 2006

Starting with a(0)=0 would give -A007526(n); starting with a(0)=1 would give -A038156(n). In general, for this recurrence, a(n) = ceiling(1 + n!*(a(0)-e)) for n>0; this is the first case with positive terms.

			a(5) = 5*(a(4)-1) = 5*(8-1) = 35.

a=3;Table[a=a*n-n,{n,1,2*4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2010 *)
RecurrenceTable[{a[0]==3,a[n]==n(a[n-1]-1)},a,{n,20}] (* Harvey P. Dale, Jul 17 2018 *)

a(n) = ceiling(1 + n!*(3-e)) for n>0.

a(n) = n! - floor(e*n!) + 1, n>0. - Gary Detlefs, Jun 06 2010

A371686 Triangle read by rows: T(n, k) = e * binomial(n, k) * Gamma(k + 1, 1).

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 1, 6, 15, 16, 1, 8, 30, 64, 65, 1, 10, 50, 160, 325, 326, 1, 12, 75, 320, 975, 1956, 1957, 1, 14, 105, 560, 2275, 6846, 13699, 13700, 1, 16, 140, 896, 4550, 18256, 54796, 109600, 109601, 1, 18, 180, 1344, 8190, 41076, 164388, 493200, 986409, 986410

Offset: 0

Peter Luschny, Apr 03 2024

			Triangle starts:
  [0] 1;
  [1] 1,  2;
  [2] 1,  4,   5;
  [3] 1,  6,  15,  16;
  [4] 1,  8,  30,  64,   65;
  [5] 1, 10,  50, 160,  325,  326;
  [6] 1, 12,  75, 320,  975, 1956,  1957;
  [7] 1, 14, 105, 560, 2275, 6846, 13699, 13700;

Cf. A000522 (main diagonal), A007526 (subdiagonal), A010842 (row sums), A000142 and A133942 (alternating row sums), A367963 (central terms).

T := (n, k) -> binomial(n, k)*GAMMA(k + 1, 1)*exp(1):
seq(seq(simplify(T(n, k)), k = 0..n), n = 0..9);

T[n_,k_]:=(n!/(n-k)!)*Sum[1/j!,{j,0,k}];Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Apr 06 2024 *)

T(n, k) = (n! / (n - k)!)*(Sum_{j = 0..k} (1 / j!)). - Detlef Meya, Apr 06 2024

A374844 a(n) = n! * Sum_{k=1..n} k^k / k!.

Original entry on oeis.org

0, 1, 6, 45, 436, 5305, 78486, 1372945, 27760776, 637267473, 16372674730, 465411092641, 14501033559948, 491388542871577, 17991446425760094, 707765586767260785, 29770993461985724176, 1333347150740094075169, 63346656788618230928466

Offset: 0

Seiichi Manyama, Jul 22 2024

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A007526, A030297, A337001, A337002.

Cf. A000142, A277506.

a:= proc(n) a(n):= n*a(n-1) + n^n end: a(0):= 0:
seq(a(n), n=0..23);  # Alois P. Heinz, Jul 22 2024

PARI
```
a(n) = n!*sum(k=1, n, k^k/k!);
```

a(0) = 0; a(n) = n*a(n-1) + n^n.
a(n) = A277506(n) - n!.
E.g.f.: -1/( (1 + 1/LambertW(-x)) * (1 - x) ).
a(n) ~ n^n / (1 - exp(-1)). - Vaclav Kotesovec, Jul 22 2024

cp's OEIS Frontend

A117643 a(n) = n*(a(n-1)-1) starting with a(0)=3.

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

Formula

A371686 Triangle read by rows: T(n, k) = e * binomial(n, k) * Gamma(k + 1, 1).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A374844 a(n) = n! * Sum_{k=1..n} k^k / k!.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula