A072364 -id:A072364 - cp's OEIS frontend

A010503 Decimal expansion of 1/sqrt(2).

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

Offset: 0

N. J. A. Sloane

The decimal expansion of sqrt(50) = 5*sqrt(2) = 7.0710678118654752440... gives essentially the same sequence.
Also real and imaginary part of the square root of the imaginary unit. - Alonso del Arte, Jan 07 2011
1/sqrt(2) = (1/2)^(1/2) = (1/4)^(1/4) (see the comments in A072364).
If a triangle has sides whose lengths form a harmonic progression in the ratio 1 : 1/(1 + d) : 1/(1 + 2d) then the triangle inequality condition requires that d be in the range -1 + 1/sqrt(2) < d < 1/sqrt(2). - Frank M Jackson, Oct 11 2011
Let s_2(n) be the sum of the base-2 digits of n and epsilon(n) = (-1)^s_2(n), the Thue-Morse sequence A010060, then Product_{n >= 0} ((2*n + 1)/(2*n + 2))^epsilon(n) = 1/sqrt(2). - Jonathan Vos Post, Jun 03 2012
The square root of 1/2 and thus it follows from the Pythagorean theorem that it is the sine of 45 degrees (and the cosine of 45 degrees). - Alonso del Arte, Sep 24 2012
Circumscribed sphere radius for a regular octahedron with unit edges. In electrical engineering, ratio of effective amplitude to peak amplitude of an alternating current/voltage. - Stanislav Sykora, Feb 10 2014
Radius of midsphere (tangent to edges) in a cube with unit edges. - Stanislav Sykora, Mar 27 2014
Positive zero of the Hermite polynomial of degree 2. - A.H.M. Smeets, Jun 02 2025

			0.7071067811865475...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Sections 1.1, 7.5.2, and 8.2, pp. 1-3, 468, 484, 487.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Harry J. Smith, Table of n, a(n) for n = 0..20000
P. C. Fishburn and J. A. Reeds, Bell inequalities, Grothendieck's constant and root two, SIAM J. Discrete Math., Vol. 7, No. 1, Feb. 1994, pp. 48-56.
Ovidiu Furdui, Problem 1, Problem Corner, Research Group in Mathematical Inequalities and Applications, 2010.
Michael Penn, A surprisingly convergent limit, YouTube video, 2022.
Michael Penn, The infinite fraction of your dreams (nightmare?), YouTube video, 2022.
Jonathan Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; arXiv:1108.6096 [math.NT], 2011, see p. 3 in the link.
Eric Weisstein's World of Mathematics, Digit Product.
Wikipedia, Platonic solid.
Donald R. Woods, Problem E 2692, Elementary Problems, The American Mathematical Monthly, Vol. 85, No. 1 (1978), p. 48; A Transcendental Function Satisfy a Duplication Formula, by David Robbins, ibid., Vol. 86, No. 5 (1979), pp. 394-395.
Index entries for algebraic numbers, degree 2.

Cf. A000120, A040042, A072364, A268682.
Cf. A073084 (infinite tetration limit).
Platonic solids circumradii: A010527 (cube), A019881 (icosahedron), A179296 (dodecahedron), A187110 (tetrahedron).
Platonic solids midradii: A020765 (tetrahedron), A020761 (octahedron), A019863 (icosahedron), A239798 (dodecahedron).
Cf. A000217, A010060, A019812, A019824, A019851, A019857, A019884, A019896.

1/Sqrt(2); // Vincenzo Librandi, Feb 21 2016

Digits:=100; evalf(1/sqrt(2)); Wesley Ivan Hurt, Mar 27 2014

N[ 1/Sqrt[2], 200]
RealDigits[1/Sqrt[2],10,120][[1]] (* Harvey P. Dale, Mar 25 2019 *)

default(realprecision, 20080); x=10*(1/sqrt(2)); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b010503.txt", n, " ", d)); \\ Harry J. Smith, Jun 02 2009

1/sqrt(2) = cos(Pi/4) = sqrt(2)/2. - Eric Desbiaux, Nov 05 2008
a(n) = 9 - A268682(n). As constants, this sequence is 1 - A268682. - Philippe Deléham, Feb 21 2016
From Amiram Eldar, Jun 29 2020: (Start)
Equals sin(Pi/4) = cos(Pi/4).
Equals Integral_{x=0..Pi/4} cos(x) dx. (End)
Equals (1/2)*A019884 + A019824 * A010527 = A019851 * A019896 + A019812 * A019857. - R. J. Mathar, Jan 27 2021
Equals hypergeom([-1/2, -3/4], [5/4], -1). - Peter Bala, Mar 02 2022
Limit_{n->oo} (sqrt(T(n+1)) - sqrt(T(n))) = 1/sqrt(2), where T(n) = n(n+1)/2 = A000217(n) is the triangular numbers. - Jules Beauchamp, Sep 18 2022
Equals Product_{k>=0} ((2*k+1)/(2*k+2))^((-1)^A000120(k)) (Woods, 1978). - Amiram Eldar, Feb 04 2024
From Stefano Spezia, Oct 15 2024: (Start)
Equals 1 + Sum_{k>=1} (-1)^k*binomial(2*k,k)/2^(2*k) [Newton].
Equal Product_{k>=1} 1 - 1/(4*(2*k - 1)^2). (End)
Equals Product_{k>=0} (1 - (-1)^k/(6*k+3)). - Amiram Eldar, Nov 22 2024

More terms from Harry J. Smith, Jun 02 2009

A073226 Decimal expansion of e^e.

Original entry on oeis.org

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

Offset: 2

Rick L. Shepherd, Jul 21 2002

Given z > 0, there exist positive real numbers x < y, with x^y = y^x = z, if and only if z > e^e. In that case, 1 < x < e < y and (x, y) = ((1 + 1/t)^t, (1 + 1/t)^(t+1)) for some t > 0. (For example, t = 1 gives 2^4 = 4^2 = 16 > e^e.) Proofs of these classical results and applications of them are in Marques and Sondow (2010).
e^e = lim_{n->infinity} ((n+1)/n)^((n+1)^(n+1)/n^n), n > 0 an integer; cf. [Vernescu] wherein it is also stated that the assertions of the previous comment above were proved by Alexandru Lupas in 2006. - L. Edson Jeffery, Sep 18 2012
A weak form of Schanuel's Conjecture implies that e^e is transcendental--see Marques and Sondow (2012).

			15.15426224147926418976043027262991190552854853685613976914...

Harry J. Smith, Table of n, a(n) for n = 2..20000
D. Marques and J. Sondow, The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture, arXiv:1212.6931 [math.NT], 2012-2013.
Simon Plouffe, exp(E) to 2000 places
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164.
A. Vernescu, About the use of a result of Professor Alexandru Lupas to obtain some properties in the theory of the number e, Gen. Math., Vol. 15, No. 1 (2007), 75-80.

Cf. A073233 (Pi^Pi), A049006 (i^i), A001113 (e), A073227 (e^e^e), A004002 (Benford numbers), A056072 (floor(e^e^...^e), n e's), A072364 ((1/e)^(1/e)), A030178 (limit of (1/e)^(1/e)^...^(1/e)), A073229 (e^(1/e)), A073230 ((1/e)^e).

Exp(Exp(1)); // G. C. Greubel, May 29 2018

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

{ default(realprecision, 20080); x=exp(1)^exp(1)/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b073226.txt", n, " ", d)); } \\ Harry J. Smith, Apr 30 2009

Equals Sum_{n>=0} e^n/n!. - Richard R. Forberg, Dec 29 2013

Equals Product_{n>=0} e^(1/n!). - Amiram Eldar, Jun 29 2020

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

Original entry on oeis.org

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

A083648 Decimal expansion of Sum_{n>=1} -(-1)^n/n^n = Integral_{x=0..1} x^x dx.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, May 01 2003

In 1697, Johann Bernoulli explores this curve and finds its minimum and the area under the curve from 0 to 1, all this without the benefit of the exponential function.

			0.78343051071213440705926438652697546940768199014693095825541782270...

William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton, NJ 2005, pp. 46-51.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.
Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1), Princeton, New Jersey: Princeton University Press (1988), p. 146.

G. C. Greubel, Table of n, a(n) for n = 0..5000
M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
Eric Weisstein's World of Mathematics, Power Tower.
Eric Weisstein's World of Mathematics, Sophomore's Dream.

Cf. A137420 (continued fraction expansion).

Cf. A073009. The minimum point on the curve x^x is (A068985, A072364).

RealDigits[ Sum[ -(-1)^n /n^n, {n, 1, 60}], 10, 111] [[1]] (* Robert G. Wilson v, Jan 31 2005 *)

-sumalt(n=1, (-1/n)^(n)) \\ Michel Marcus, Oct 15 2015

numerical_approx(-sum((-1/n)^n for n in (1..120)), digits=130) # G. C. Greubel, Mar 01 2019

Constant also equals the double integral Integral_{y = 0..1} Integral_{x = 0..1} (x*y)^(x*y) dx dy. - Peter Bala, Mar 04 2012
From Petros Hadjicostas, Jun 29 2020: (Start)
Equals -Integral_{x=0..1, y=0..1} (x*y)^(x*y)/log(x*y) dx dy. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to Integral_{x=0..1} x^x dx.)
Equals -Integral_{x=0..1} x^x*log(x) dx. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to the double integral of Peter Bala above.)
Without using the results in Glasser (2019), notice that Integral x^x*(1 + log(x)) dx = x^x + c, which implies Integral_{x=0..1} x^x dx = -Integral_{x=0..1} x^x*log(x) dx. (End)

A073230 Decimal expansion of (1/e)^e.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 22 2002

(1/e)^e = e^(-e) = 1/(e^e) (reciprocal of A073226).

The power tower function f(x)=x^(x^(x^...)) is defined on the closed interval [e^(-e),e^(1/e)]. - Lekraj Beedassy, Mar 17 2005

			0.06598803584531253707679018759...

Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, Solution to problem 8A (Power Tower) p. 240.

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; see the Appendix.

Cf. A001113 (e), A068985 (1/e), A073229 (e^(1/e)), A072364 ((1/e)^(1/e)), A073226 (e^e).

Exp(-1)^Exp(1); // G. C. Greubel, May 29 2018

a=IntegerDigits[IntegerPart[(1/E)^E*10^99]];PrependTo[a,0] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2008 *)
RealDigits[(1/E)^E, 10, 100][[1]] (* Alonso del Arte, Aug 26 2011 *)

PARI
```
exp(-1)^exp(1)
```

A073232 Decimal expansion of ((1/e)^(1/e))^(1/e).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 22 2002

			0.87342301849311664298903234866...

Cf. A001113 (e), A068985 (1/e), A072364 ((1/e)^(1/e)), A073231 ((1/e)^(1/e)^(1/e)), A073228 ((e^e)^e), A073227 (e^e^e).

RealDigits[((1/E)^(1/E))^(1/E),10,120][[1]]  (* Harvey P. Dale, Apr 21 2011 *)

PARI
```
exp(-exp(-2))
```

Equals e^(-e^(-2)).

A072365 Decimal expansion of (1/3)^(1/3).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 18 2002

Minimum value of (1/n)^(1/n) for integer n>0.

			.693361274350634704843352274785...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A072364.

(1/3)^(1/3); // G. C. Greubel, May 29 2018

RealDigits[Surd[1/3,3],10,120][[1]] (* Harvey P. Dale, Nov 03 2013 *)

PARI
```
(1/3)^(1/3)
```

A073231 Decimal expansion of (1/e)^(1/e)^(1/e).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 22 2002

			0.50047350056363684054513490133...

Cf. A001113 (e), A068985 (1/e), A072364 ((1/e)^(1/e)), A030178 (limit of (1/e)^(1/e)^...^(1/e)), A073232 (((1/e)^(1/e))^(1/e)), A073227 (e^e^e).

With[{c=1/E},RealDigits[c^c^c,10,120][[1]]] (* Harvey P. Dale, Jul 16 2025 *)

PARI
```
exp(-1)^exp(-1)^exp(-1)
```

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

A194624 Decimal expansion of the smaller solution to x^x = 3/4.

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Sep 02 2011

Since (1/e)^(1/e) < 3/4 < 1, the equation x^x = 3/4 has two solutions x = a and x = b with 0 < a < 1/e < b < 1. Both solutions are transcendental (see Proposition 2.2 in Sondow-Marques 2010).

			0.15351678966395294715006833297846322771126948548996962031798542833437261364190...

J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164.
Index entries for transcendental numbers

Cf. A030798 (x^x = 2), A072364 ((1/e)^(1/e)), A194625 (larger solution to x^x = 3/4).

x = x /. FindRoot[x^x == 3/4, {x, 0.1}, WorkingPrecision -> 120]; RealDigits[x, 10, 100] // First

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A010503 Decimal expansion of 1/sqrt(2).

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A073226 Decimal expansion of e^e.

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

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A083648 Decimal expansion of Sum_{n>=1} -(-1)^n/n^n = Integral_{x=0..1} x^x dx.

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Mathematica

PARI

Sage

Formula

A073230 Decimal expansion of (1/e)^e.

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Magma

Mathematica

PARI

A073232 Decimal expansion of ((1/e)^(1/e))^(1/e).

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Mathematica