A068985 -id:A068985 - cp's OEIS frontend

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

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)).

A087654 Decimal expansion of D(1) where D(x) is the Dawson function.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Sep 25 2003

			0.5380795069127684191363874204075567547919750017539...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 42, page 407.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Dawson's Integral.

RealDigits[ N[ Sqrt[Pi]*Erfi[1]/(2*E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)
RealDigits[DawsonF[1], 10, 120][[1]] (* Amiram Eldar, Jun 25 2023 *)

intnum(t=0, 1, exp(t^2))/exp(1) \\ Michel Marcus, Feb 28 2023

D(1) = (1/e)*Integral_{t=0..1} exp(t^2) dt.
Equals Integral_{x=0..oo} e^(-x^2) sin(2x) dx = 1F1(1;3/2;-1). - R. J. Mathar, Jul 10 2024
Equals A099288 * sqrt(Pi)/(2e) = A099288 *A019704 * A068985. - R. J. Mathar, Jul 10 2024

A092615 Decimal expansion of e^(-1/3).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

			0.71653131057378925042560409692537966745311205982147...

Cf. A001113, A019774, A068985, A092041 (reciprocal).

RealDigits[E^(-1/3),10,120][[1]] (* Harvey P. Dale, Feb 13 2012 *)

Equals lim_{x->0} (tanh(x)/x)^(1/x^2). - Amiram Eldar, Jul 04 2022

A152887 Number of descents beginning with an even number and ending with an odd number in all permutations of {1,2,...,n}.

Original entry on oeis.org

0, 1, 2, 18, 72, 720, 4320, 50400, 403200, 5443200, 54432000, 838252800, 10059033600, 174356582400, 2440992153600, 47076277248000, 753220435968000, 16005934264320000, 288106816757760000, 6690480522485760000, 133809610449715200000, 3372002183332823040000

Offset: 1

Emeric Deutsch, Jan 19 2009

nonn

a(n) is the number of ways to perform the following: Divide the set {1,2,...,n} into three pairwise disjoint subsets, A,B,C so that A union B union C = {1,2,...,n}. Let A contain an odd number of elements and B contain an even number of elements. Linearly order the elements within each subset. - Geoffrey Critzer, Sep 26 2011

			a(8) = 50400 because (i) the descent pairs can be chosen in 1+2+3+4 = 10 ways, namely (2,1), (4,1), (4,3), (6,1), (6,3), (6,5), (8,1), (8,3), (8,5), (8,7); (ii) they can be placed in 7 positions, namely (1,2), (2,3), (3,4), (4,5), (5,6), (6,7), (7,8); (iii) the remaining 6 entries can be permuted in 6! = 720 ways; 10*7*720 = 50400.

Miklos Bona, A Walk Through Combinatorics, World Scientific Publishing Co., 2002, page 170.

Vincenzo Librandi, Table of n, a(n) for n = 1..400

Cf. A152885, A152886.

Cf. A001620, A068985, A099284.

[Factorial(n-1)*(2*n*(n+1)+(2*n+1)*(-1)^n-1)/16: n in [1..20]]; // Bruno Berselli, Nov 07 2011

a := proc (n) if `mod`(n, 2) = 0 then factorial(n-1)*binomial((1/2)*n+1, 2) else factorial(n-1)*binomial((1/2)*n+1/2, 2) end if end proc: seq(a(n), n = 1 .. 22);

CoefficientList[Series[x/((1 - x) (1 - x^2)^2), {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* Geoffrey Critzer, Mar 03 2010 *)

a(2n) = (2n-1)!*C(n+1,2);  a(2n+1) = (2n)!*C(n+1,2).
E.g.f.: x/((1-x^2)^2*(1-x)). - Geoffrey Critzer, Mar 03 2010
a(n) = (n-1)!*(2*n*(n+1)+(2*n+1)*(-1)^n-1)/16. - Bruno Berselli, Nov 07 2011
D-finite with recurrence a(n) -2*a(n-1) +(-n^2+2)*a(n-2) +n*(n-3)*a(n-3)=0. - R. J. Mathar, Jul 26 2022
Sum_{n>=2} 1/a(n) = 4*(CoshIntegral(1) - gamma - 1/e) + 2 = 4*(A099284 - A001620 - A068985) + 2. - Amiram Eldar, Jan 22 2023

A246665 Decimal expansion of the asymptotic probability of success in the full-information secretary problem with uniform distribution when the number of applicants is also uniformly distributed.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 01 2014

In this variant of the secretary problem, the applicants' values are independently distributed on a known interval, like in A242674; and the number of applicants is itself a random variable with uniform distribution on 1..n (and then the limit n -> infinity is taken), like in A325905. So we have more information than in the variant considered in A325905 but less information than in the variant considered in A242674. Hence A325905 < this constant < A242674. - Andrey Zabolotskiy, Sep 14 2019

			0.43517080558012765805918991284785841042796259475347024702979123...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.15 Optimal stopping constants, p. 361.

Steven R. Finch, Errata and Addenda to Mathematical Constants, Sec. 5.15 Optimal Stopping Constants, pp. 53-55, arXiv:2001.00578 [math.HO], 2020.
Zdzisław Porosiński, On best choice problems having similar solutions, Statistics & Probability Letters, 56 (2002), 321-327.
Eric Weisstein's MathWorld, Sultan's Dowry Problem
Wikipedia, Secretary problem

Cf. A246664, A068985, A325905, A242674.

a = x /. FindRoot[E^x*(1 - EulerGamma - Log[x] + ExpIntegralEi[-x]) - (EulerGamma + Log[x] - ExpIntegralEi[x]) == 1, {x, 2}, WorkingPrecision -> 102]; (1 - E^a)*ExpIntegralEi[-a] - (E^-a + a*ExpIntegralEi[-a])*(EulerGamma + Log[a] - ExpIntegralEi[a]) // RealDigits // First

(1 - e^a)*Ei(-a) - (e^(-a) + a*Ei(-a))*(gamma + log(a) - Ei(a)), where a is A246664, gamma is Euler's constant and Ei is the exponential integral function.

A300916 Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/e.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Mar 16 2018

			0.557919282255416046773864733137284325268059522147000568856861678665669168...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Tetration

Cf. A001113, A030178, A068985, A277522, A277651, A293009.

RealDigits[E*Exp[-2*LambertW[1]]/(1+LambertW[1]), 10, 100][[1]] (* G. C. Greubel, Sep 09 2018 *)

exp(1)*exp(-2*lambertw(1))/(1+lambertw(1)) \\ Michel Marcus, Mar 16 2018

Equals exp(1)*exp(-2*LambertW(1))/(1+LambertW(1)).

A052521 Number of pairs of sequences of cardinality at least 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 10080, 120960, 1451520, 18144000, 239500800, 3353011200, 49816166400, 784604620800, 13076743680000, 230150688768000, 4268249137152000, 83230858174464000, 1703031405723648000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..445
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 88.

Cf. sequences with formula (n + k)*n! listed in A282466.

Cf. A001113, A001620, A068985, A091725, A099285.

Concatenation([0,0,0,0,0,0], List([6..20], n-> (n-5)*Factorial(n))); # G. C. Greubel, May 13 2019

[n le 5 select 0 else (n-5)*Factorial(n): n in [0..20]]; // G. C. Greubel, May 13 2019

spec := [S,{B=Sequence(Z,3 <= card), S=Prod(B,B)},labeled]: # Pairs spec
seq(combstruct[count](spec, size=n), n=0..20);

Table[If[n<6, 0, (n-5)*n!], {n,0,20}] (* G. C. Greubel, May 13 2019 *)

{a(n) = if(n<6, 0, (n-5)*n!)}; \\ G. C. Greubel, May 13 2019

[0,0,0,0,0,0]+[(n-5)*factorial(n) for n in (6..20)] # G. C. Greubel, May 13 2019

E.g.f.: x^6/(1-x)^2.
(n-5)*a(n+1) + (4 + 3*n - n^2)*a(n) = 0, with a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = 0, a(6) = 720.
a(n) = (n-5)*n!.
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=6} 1/a(n) = 5477/7200 - 17*e/60 - gamma/120 + Ei(1)/120 = 5477/7200 - (17/60)*A001113 - (1/120)*A001620 + A091725/120.
Sum_{n>=6} (-1)^n/a(n) = 403/7200 - 1/(6*e) + gamma/120 - Ei(-1)/120 = 403/7200 - (1/6)*A068985 + (1/120)*A001620 + (1/120)*A099285. (End)

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)
```

A092554 Decimal expansion of e^(-3).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 09 2004

			0.049787068367863942979342415650061776631699592188423...

Cf. A019774, A001113, A068985.

First[RealDigits[E^-3, 10, 100, -1]] (* Paolo Xausa, Feb 09 2025 *)

PARI
```
exp(-3) \\ Michel Marcus, May 10 2022
```

A092555 Decimal expansion of e^(-4).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 09 2004

This is one of Rényi's parking constants. - Alonso del Arte, Dec 28 2013

			0.0183156388887...

Steven R. Finch, Mathematical Constants, Cambridge University Press (2003): 280.

Cf. A019774, A001113, A068985, A092553 (e^(-2)), A092554 (e^(-3)).

RealDigits[E^(-4), 10, 100][[1]] (* Alonso del Arte, Dec 28 2013 *)

cp's OEIS Frontend

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

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A087654 Decimal expansion of D(1) where D(x) is the Dawson function.

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A092615 Decimal expansion of e^(-1/3).

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A152887 Number of descents beginning with an even number and ending with an odd number in all permutations of {1,2,...,n}.

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A246665 Decimal expansion of the asymptotic probability of success in the full-information secretary problem with uniform distribution when the number of applicants is also uniformly distributed.

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A300916 Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/e.

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A052521 Number of pairs of sequences of cardinality at least 3.

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