A332408 -id:A332408 - cp's OEIS frontend

A073229 Decimal expansion of e^(1/e).

1, 4, 4, 4, 6, 6, 7, 8, 6, 1, 0, 0, 9, 7, 6, 6, 1, 3, 3, 6, 5, 8, 3, 3, 9, 1, 0, 8, 5, 9, 6, 4, 3, 0, 2, 2, 3, 0, 5, 8, 5, 9, 5, 4, 5, 3, 2, 4, 2, 2, 5, 3, 1, 6, 5, 8, 2, 0, 5, 2, 2, 6, 6, 4, 3, 0, 3, 8, 5, 4, 9, 3, 7, 7, 1, 8, 6, 1, 4, 5, 0, 5, 5, 7, 3, 5, 8, 2, 9, 2, 3, 0, 4, 7, 0, 9, 8, 8, 5, 1, 1, 4, 2, 9, 5

Offset: 1

Rick L. Shepherd, Jul 22 2002

e^(1/e) = 1/((1/e)^(1/e)) (reciprocal of A072364).
Let w(n+1)=A^w(n); then w(n) converges if and only if (1/e)^e <= A <= e^(1/e) (see the comments in A073230) for initial value w(1)=A. If A=e^(1/e) then lim_{n->infinity} w(n) = e. - Benoit Cloitre, Aug 06 2002; corrected by Robert FERREOL, Jun 12 2015
x^(1/x) is maximum for x = e and the maximum value is e^(1/e). This gives an interesting and direct proof that 2 < e < 4 as 2^(1/2) < e^(1/e) > 4^(1/4) while 2^(1/2) = 4^(1/4). - Amarnath Murthy, Nov 26 2002
For large n, A234604(n)/A234604(n-1) converges to e^(1/e). - Richard R. Forberg, Dec 28 2013
Value of the unique base b > 0 for which the exponential curve y=b^x and its inverse y=log_b(x) kiss each other; the kissing point is (e,e). - Stanislav Sykora, May 25 2015
Actually, there is another base with such property, b=(1/e)^e with kiss point (1/e,1/e). - Yuval Paz, Dec 29 2018
The problem of finding the maximum of f(x) = x^(1/x) was posed and solved by the Swiss mathematician Jakob Steiner (1796-1863) in 1850. - Amiram Eldar, Jun 17 2021

			1.44466786100976613365833910859...

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 35.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Ovidiu Furdui, Problem 11982, The American Mathematical Monthly, Vol. 124, No. 5 (2017), p. 465; A Limit of a Power of a Sum, Solution to Problem 11982 by Roberto Tauraso, ibid., Vol. 126, No. 2 (2019), p. 187.
Simon Plouffe, exp(1/e).
Jonathan Sondow and Diego Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae, Vol. 37 (2010), pp. 151-164; see Definition 4.1 on p. 158.
Jacob Steiner, Über das größte Product der Theile oder Summanden jeder Zahl, Crelle, Vol. 40 (1850), pp. 208; alternative link.
Eric Weisstein's World of Mathematics, Steiner's Problem.

Cf. A001113 (e), A068985 (1/e), A073230 ((1/e)^e), A072364 ((1/e)^(1/e)), A073226 (e^e).
Cf. A093157, A103476.
Cf. A038051, A086331, A252782, A270593, A270917, A270923, A277473, A309652, A332408.

evalf[110](exp(exp(-1))); # Muniru A Asiru, Dec 29 2018

Mathematica
```
RealDigits[ E^(1/E), 10, 110] [[1]]
```
PARI
```
exp(1)^exp(-1)
```

Equals 1 + Integral_{x = 1/e..1} (1 + log(x))/x^x dx = 1 - Integral_{x = 0..1/e} (1 + log(x))/x^x dx. - Peter Bala, Oct 30 2019
Equals Sum_{k>=0} exp(-k)/k!. - Amiram Eldar, Aug 13 2020
Equals lim_{x->oo} (Sum_{n>=1} (x/n)^n)^(1/x) (Furdui, 2017). - Amiram Eldar, Mar 26 2022

A351765 a(n) = n! * Sum_{k=0..n} n^(n-k) * (n-k)^k/k!.

Original entry on oeis.org

1, 1, 12, 279, 11536, 746525, 69768036, 8902181575, 1487939919936, 315597946293657, 82839437215344100, 26366747854082944451, 10006618140321691249296, 4464690010732922712332149, 2313871692128866349730705924, 1378552938661073773617331110975

Offset: 0

Seiichi Manyama, Feb 18 2022

nonn

Main diagonal of A351761.

Cf. A332408, A351768, A351779.

Join[{1}, Table[n!*Sum[n^(n - k)*(n - k)^k/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Feb 19 2022 *)

a(n) = n!*sum(k=0, n, n^(n-k)*(n-k)^k/k!);

a(n) = n! * [x^n] 1/(1 - n*x*exp(x)).
From Vaclav Kotesovec, Feb 19 2022: (Start)
a(n) ~ exp(1) * n! * n^n.
a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n-1). (End)

A332627 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * k! * k^n.

Original entry on oeis.org

1, 1, 6, 117, 4388, 266065, 23731314, 2923345621, 475364541672, 98623225721601, 25421365316232710, 7969388199705535141, 2985785305877403047820, 1317500933136749853197329, 676266417871227455138941242, 399516621958550611386236160405

Offset: 0

Ilya Gutkovskiy, Apr 23 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A000166, A120485, A134095, A302397, A319392, A332408.

Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] k! k^n, {k, 0, n}], {n, 1, 15}]]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * k! * k^n); \\ Michel Marcus, Apr 24 2020

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x*exp(-x))^k))) \\ Seiichi Manyama, Feb 19 2022

G.f.: Sum_{k>=0} k! * k^k * x^k / (1 + k*x)^(k+1).
a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^n / (n-k)!.
a(n) ~ c * n! * n^n, where c = A072364 = exp(-exp(-1)). - Vaclav Kotesovec, Jul 10 2021
E.g.f.: Sum_{k>=0} (k*x*exp(-x))^k. - Seiichi Manyama, Feb 19 2022

A341185 a(n) = Sum_{k=0..n} k^n * k! * binomial(n,k)^2.

Original entry on oeis.org

1, 1, 12, 315, 15088, 1141625, 124989156, 18659050795, 3638892086208, 897534389449809, 272981684150035300, 100316132701760094251, 43802068733570039425776, 22409162143775383385763913, 13274030650412266312507931652

Offset: 0

Seiichi Manyama, Feb 06 2021

nonn

Cf. A002720, A332408, A336828, A341197.

a[0] = 1; a[n_] := Sum[k^n * k! * Binomial[n, k]^2, {k, 0, n}]; Array[a, 15, 0] (* Amiram Eldar, Feb 06 2021 *)

a(n) = sum(k=0, n, k^n*k!*binomial(n, k)^2);

cp's OEIS Frontend

A073229 Decimal expansion of e^(1/e).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A351765 a(n) = n! * Sum_{k=0..n} n^(n-k) * (n-k)^k/k!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A332627 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * k! * k^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A341185 a(n) = Sum_{k=0..n} k^n * k! * binomial(n,k)^2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI