A135002 -id:A135002 - cp's OEIS frontend

A112302 Decimal expansion of quadratic recurrence constant sqrt(1 * sqrt(2 * sqrt(3 * sqrt(4 * ...)))).

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

Offset: 1

Michael Somos, Sep 02 2005

From Johannes W. Meijer, Jun 27 2016: (Start)
With Phi(z, p, q) the Lerch transcendent, define LP(n) = (1/n) * sum(Phi(1/2, n-k, 1) * LP(k), k=0..n-1), with LP(0) = 1. Conjecture: Lim_{n -> infinity} LP(n) = A112302.
For similar formulas, see A090998 and A135002. For background information, see A274181.
The structure of the n! * LP(n) formulas leads to the multinomial coefficients A036039. (End)

			1.6616879496335941212958189227499507499644186350250682081897111680...

S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., AMS Chelsea 2000. See Appendix I. p. 348.

Robert G. Wilson v, Table of n, a(n) for n = 1..1011
Steven Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, Section 6.10.
Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, On integrality and asymptotic behavior of the (k,l)-Göbel sequences, arXiv:2402.09064 [math.NT], 2024. See p. 2.
Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.
M. D. Hirschhorn, A note on Somos' quadratic recurrence constant, J. Number Theory 131 (2011), 2061-2063.
Dawei Lu and Zexi Song, Some new continued fraction estimates of the Somos' quadratic recurrence constant, Journal of Number Theory, 155 (2015), 36-45.
Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant, Results in Mathematics 74(1) (2019), Article 6.
Cristinel Mortici, Estimating the Somos' quadratic recurrence constant, J. Number Theory 130 (2010), 2650-1657.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, arXiv:math/0506319 [math.NT], 2005-2006; see page 8.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008), 247-270.
Jörg Neunhäuserer, On the universality of Somos' constant, arXiv:2006.02882 [math.DS], 2020.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007), 292-314.
Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.
Wikipedia, Somos' quadratic recurrence constant
Xu You and Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, 436(1) (2016), 513-520.

Cf. A052129, A055209, A116603, A123851, A123852, A123853, A123854.
Cf. A114124 (log).
Cf. A036039, A090998, A135002, A188834, A259235, A274181.

RealDigits[ Fold[ N[ Sqrt[ #2*#1], 128] &, Sqrt@ 351, Reverse@ Range@ 350], 10, 111][[1]] (* Robert G. Wilson v, Nov 05 2010 *)
Exp[-Derivative[1, 0][PolyLog][0, 1/2]] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Apr 07 2014, after Jonathan Sondow *)

{a(n) = if( n<-1, 0, n++; default( realprecision, n+2); floor( prodinf( k=1, k^2^-k)* 10^n) % 10)};

prodinf(n=1,n^2^-n) \\ Charles R Greathouse IV, Apr 07 2013

from mpmath import polylog, diff, exp, mp
mp.dps = 120
somos_const = exp(-diff(lambda n: polylog(n, 1/2), 0))
A112302 = [int(d) for d in mp.nstr(somos_const, n=mp.dps)[:-1] if d != '.']  # Jwalin Bhatt, Nov 23 2024

Equals Product_{n>=1} n^(1/2^n). - Jonathan Sondow, Apr 07 2013
Equals exp(A114124) = A188834/2 = sqrt(A259235). - Hugo Pfoertner, Nov 23 2024
From Jwalin Bhatt, Apr 02 2025: (Start)
Equals exp(-PolyLog'(0,1/2)), where PolyLog'(x,y) represents the derivative of the polylogarithm w.r.t. x.
Equals Product_{n>=1} (1+1/n)^(1/2^n).
Equals exp(Sum_{n>=2} log(n)/2^n).
Equals 2*exp(Sum_{n>=1} (log(1+1/n)-1/n)/2^n). (End)

A244009 Decimal expansion of 1 - log(2).

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Jun 17 2014

Fraction of numbers which are sqrt-smooth, see A048098 and A063539. - Charles R Greathouse IV, Jul 14 2014

Asymptotic survival probability in the 100 prisoners problem. - Alois P. Heinz, Jul 08 2022

			0.30685281944005469058276787854...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Peter Bala, A continued fraction expansion for the constant 1 - log(2)
Donald E. Knuth and Luis Trabb Pardo, Analysis of a simple factorization algorithm, Theoretical Computer Science 3:3 (1976), pp. 321-348.
The Ramanujan Machine, Using algorithms to discover new mathematics.
Wikipedia, 100 prisoners problem
Index entries for transcendental numbers

Cf. A002162, A019739, A024168, A100381, A135002, A188859.

Essentially the same digits as A239354.

f:= sum(1/(2*k*(2*k+1)), k=1..infinity):
s:= convert(evalf(f, 140), string):
seq(parse(s[i+1]), i=1..106);  # Alois P. Heinz, Jun 17 2014

RealDigits[1-Log[2],10,120][[1]] (* Harvey P. Dale, Sep 23 2016 *)

1-log(2) \\ Charles R Greathouse IV, Jul 14 2014

Equals Sum_{k>=0} 1/(2*k*(2*k+1)) = A239354 + 1/4 = A188859/2.
From Amiram Eldar, Aug 07 2020: (Start)
Equals Sum_{k>=1} 1/(k*(k+1)*2^k) = Sum_{k>=2} 1/A100381(k).
Equals Sum_{k>=2} (-1)^k * zeta(k)/2^k.
Equals Integral_{x=1..oo} 1/(x^2 + x^3) dx. (End)
Equals log(e/2) = log(A019739) = -log(2/e) = -log(A135002). - Wolfdieter Lang, Mar 04 2022
Equals lim_{n->oo} A024168(n)/n!. - Alois P. Heinz, Jul 08 2022
Equals 1/(4 - 4/(7 - 12/(10 - ... - 2*n*(n-1)/((3*n+1) - ...)))) (an equivalent continued fraction for 1 - log(2) was conjectured by the Ramanujan machine). - Peter Bala, Mar 04 2024
Equals Sum_{k>=1} zeta(2*k)/((2*k + 1)*2^(2*k-1)) (see Finch). - Stefano Spezia, Nov 02 2024

A182987 Least a + b such that a*b = A002110(n), the product of the first n primes, where a, b are positive integers.

Original entry on oeis.org

2, 3, 5, 11, 29, 97, 347, 1429, 6229, 29873, 160879, 895681, 5448239, 34885673, 228759799, 1568299433, 11417382973, 87698582693, 684947829299, 5606539600699, 47241542381273, 403631914511993, 3587558929043927, 32684217334524347, 308342289648328511, 3036819365023723883

Offset: 0

Risto Kauppila, Feb 06 2011

nonn

Original definition (not applicable for n = 0 and 1, but equivalent for n >= 2):
Let p(S) be product of integers in S. a(n) is minimum of p(S_1) + p(S_2) over all partitions of first n primes into sets S_1 and S_2.
Also: Least integer such that a(n)^2 - 4*A002110(n) is a square. - David Broadhurst, Sep 20 2011
The integers a,b are the two median divisors of primorial(n), a = A060795(n) = A060775(A002110(n)) and b = A060796(n) = A033677(A002110(n)). (For n = 0, a = b = 1 of course.) - M. F. Hasler, Sep 20 2011

			a(3) = 11 = min{ 2*3 + 5 = 11, 2*5 + 3 = 13, 3*5 + 2 = 17 }
Or, a(3) = 11 = min { 1+30, 2+15, 3+10, 5+6 } because A002110(3) = 2*3*5 = 30 = 2*15 = 3*10 = 5*6.

Max Alekseyev, Table of n, a(n) for n = 0..70
David Broadhurst, Re: adding to prime number [primes in A182987], primenumbers group, Sep 20 2011.
David Broadhurst and others, Adding to prime number, digest of 28 messages in primenumbers Yahoo group, Sep 19, 2011 - Sep 22, 2011.

Cf. A173631, A060795, A060796, A061060, A127180.

Cf. A000196 (integer sqrt), A002110 (primorial), A010052 (is_square).

a[0] = 2; a[n_] := Module[{m = Times @@ Prime[Range[n]]}, For[an = 2 Floor[Sqrt[m]] + 1, an <= m + 2, an += 2, If[IntegerQ[Sqrt[an^2 - 4 m]], Return[an]]]]; Table[an = a[n]; Print[an]; an, {n, 0, 25}] (* Jean-François Alcover, Oct 20 2016, adapted from PARI *)

A182987(n)={if(n,vecsum(divisors(vecprod(primes(n)))[2^(n-1)..2^(n-1)+1]),2)}  \\ Needs stack size >= 2^(n+6). - M. F. Hasler, Sep 20 2011, edited Mar 24 2022

A182987(n)={ n||return(2); my(m=prod(k=1,n,prime(k))); forstep(a=2*sqrtint(m)+1,m+2,2, issquare(a^2-4*m) & return(a)) }  \\ Slow for n >= 28, but needs not much memory. - M. F. Hasler, following an idea from David Broadhurst, Sep 20 2011

def A182987(n): # becomes slow for n >= 28, but not slower than memory-hungry
   # sum(divisors(primorial(n))[2**(n-1)-1:2**(n-1)+1]) if n else 2
   "Compute A182987(n) = sum of the two central divisors of primorial(n)."
   if n < 2: return n+2
   from math import isqrt # = A000196
   from sympy import primorial # = A002110
   from sympy.ntheory.primetest import is_square # = A010052
   m = primorial(n)*4; a = isqrt(m)|1  ### ceil(sqrt(m))**2 < m  for n = 26..27 !!
   while True:
      if is_square(a*a - m): return a
      a += 2
# M. F. Hasler, Mar 21 2022

a(n) = A060795(n) + A060796(n). - M. F. Hasler, Sep 20 2011

Conjecture: Limit_{N->oo} (Sum_{n=1..N} log(a(n)-2*sqrt(prime(n)#))) / (Sum_{n=1..N} prime(n)) = 2/e - 1/2 (i.e., A135002 - 1/2). - Alain Rocchelli, Nov 30 2023

First term and example corrected, as empty sets have product 1, by Phil Carmody, Sep 20 2011
Simpler definition and extension to n = 0 by M. F. Hasler, Sep 20 2011
b-file extended to a(59) with results from R. Gerbicz (cf. 2nd Broadhurst link) by M. F. Hasler, Mar 22 2022
a(60)-a(70) in b-file from Max Alekseyev, Apr 20 2022

A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 1, 10, 478, 68248, 21809656, 13107532816, 13244650672240, 20818058883902848, 48069880140604832128, 156044927762422185270016, 687740710497308621254625536, 4000181720339888446834235653120, 29991260979682976913756629498334208

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The multinomial transform [MNL] transforms an input sequence b(n) into the output sequence a(n). Given the fact that the structure of the a(n) formulas, see the examples, lead to the multinomial coefficients A036039 the MNL transform seems to be an appropriate name for this transform. The multinomial transform is related to the exponential transform, see A274804 and the third formula. For the inverse multinomial transform [IML] see A274844.
To preserve the identity IML[MNL[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A036039 appear.
We observe that a(0) = 1 and that this term provides no information about any value of b(n), this notwithstanding we will start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the multinomial transform of a sequence. The first program uses the first formula which was found by Paul D. Hanna, see A158876, and Vladimir Kruchinin, see A215915. The second program uses properties of the e.g.f., see the sequences A158876, A213507, A244430 and A274539 and the third formula. The third program uses information about the inverse multinomial transform, see A274844.
Some MNL transform pairs are, n >= 1: A000045(n) and A244430(n-1); A000045(n+1) and A213527(n-1); A000108(n) and A213507(n-1); A000108(n-1) and A243953(n-1); A000142(n) and A158876(n-1); A000203(n) and A053529(n-1); A000110(n) and A274539(n-1); A000041(n) and A215915(n-1); A000035(n-1) and A177145(n-1); A179184(n) and A038205(n-1); A267936(n) and A000266(n-1); A267871(n) and A000090(n-1); A193356(n) and A088009(n-1).

			Some a(n) formulas, see A036039:
  a(0) = 1
  a(1) = 1*x(1)
  a(2) = 1*x(2) + 1*x(1)^2
  a(3) = 2*x(3) + 3*x(1)*x(2) + 1*x(1)^3
  a(4) = 6*x(4) + 8*x(1)*x(3) + 3*x(2)^2 + 6*x(1)^2*x(2) + 1*x(1)^4
  a(5) = 24*x(5) + 30*x(1)*x(4) + 20*x(2)*x(3) + 20*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 10*x(1)^3*x(2) + 1*x(1)^5

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, arXiv:math/0205301 [math.CO], 2002; Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274844, A274804, A036039, A001818, A001316, A215915, A008279.
Cf. A244430, A213527, A213507, A243953, A158876, A274539, A215915.
Cf. A090998, A008298, A271703, A112302, A135002, A274540.

nmax:= 13: b := proc(n): (doublefactorial(2*n-1))^2 end: a:= proc(n) option remember: if n=0 then 1 else add(((n-1)!/(n-k)!) * b(k) * a(n-k), k=1..n) fi: end: seq(a(n), n = 0..nmax); # End first MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := exp(add(b(n)*x^n/n, n = 1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n = 0..nmax); # End second MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(log(1+add(s(n)*x^n/n!, n=1..nmax)), x, nmax+1): d := proc(n): n*coeff(f, x, n) end: a(0) := 1: a(1) := b(1): s(1) := b(1): for n from 2 to nmax do s(n) := solve(d(n)-b(n), s(n)): a(n):=s(n): od: seq(a(n), n=0..nmax); # End third MNL program.

b[n_] := (2*n - 1)!!^2;
a[0] = 1; a[n_] := a[n] = Sum[((n-1)!/(n-k)!)*b[k]*a[n-k], {k, 1, n}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 17 2017 *)

a(n) = Sum_{k=1..n} ((n-1)!/(n-k)!)*b(k)*a(n-k), n >= 1 and a(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
a(n) = n!*P(n), with P(n) = (1/n)*(Sum_{k=0..n-1} b(n-k)*P(k)), n >= 1 and P(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n) with b(n) = A001818(n) = ((2*n-1)!!)^2.
denom(a(n)/2^n) = A001316(n); numer(a(n)/2^n) = [1, 1, 5, 239, 8531, 2726207, ...].

A274540 Decimal expansion of exp(sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Johannes W. Meijer, Jun 27 2016

Define P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k) for n >= 1, and P(0) = 1, with x(q) = C1 and x(n) = 1 for all other n. We find that C2 = lim_{n -> infinity} P(n) = exp((C1-1)/q).
The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.
Some transform pairs: C1 = A002162 (log(2)) and C2 = A135002 (2/exp(1)); C1 = A016627 (log(4)) and C2 = A135004 (4/exp(1)); C1 = A001113 (exp(1)) and C2 = A234473 (exp(exp(1)-1)).
From Peter Bala, Oct 23 2019: (Start)
The constant is irrational: Henry Cohn gives the following proof in Todd and Vishals Blog - "By the way, here's my favorite application of the tanh continued fraction: exp(sqrt(2)) is irrational.
Consider sqrt(2)*(exp(sqrt(2))-1)/(exp(sqrt(2))+1). If exp(sqrt(2)) were rational, or even in Q(sqrt(2)), then this expression would be in Q(sqrt(2)). However, it is sqrt(2)*tanh(1/sqrt(2)), and the tanh continued fraction shows that this equals [0,1,6,5,14,9,22,13,...]. If it were in Q(sqrt(2)), it would have a periodic simple continued fraction expansion, but it doesn't." (End)

			c = 4.113250378782927517173581815140304502401663943151...

G. C. Greubel, Table of n, a(n) for n = 1..10000
The Dev Team and Simon Plouffe, The Inverse Symbolic Calculator (ISC).
Todd Trimble and Vishal Lama, Continued fraction for e, Todd and Vishal’s blog 2008/08/04
Index entries for transcendental numbers

Cf. A274541, A274542, A036039, A135002, A135004, A234473.

Cf. A014176, A002193, A010503, A131594, A020765, A010466, A020775.

Digits := 80: evalf(exp(sqrt(2))); # End program 1.
P := proc(n) : if n=0 then 1 else P(n) := expand((1/n)*(add(x(n-k)*P(k), k=0..n-1))) fi; end: x := proc(n): if n=1 then (1 + sqrt(2)) else 1 fi: end: Digits := 49; evalf(P(120)); # End program 2.

First@ RealDigits@ N[Exp[Sqrt@ 2], 80] (* Michael De Vlieger, Jun 27 2016 *)

my(x=exp(sqrt(2))); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", ")) \\ Felix Fröhlich, Jun 27 2016

c = exp(sqrt(2)).

c = lim_{n -> infinity} P(n) with P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k) for n >= 1, and P(0) = 1, with x(1) = (1 + sqrt(2)), the silver mean A014176, and x(n) = 1 for all other n.

More terms from Jon E. Schoenfield, Mar 15 2018

A334397 Decimal expansion of (e - 2)/e.

Original entry on oeis.org

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

Offset: 0

Daniel Hoyt, Apr 26 2020

			0.2642411176571153568089524596770782651...

M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.

Cf. A001008, A001113, A002805, A019739, A135002, A309419, A334396.

RealDigits[1 - 2/E, 10, 100][[1]] (* Alonso del Arte, Apr 26 2020 *)

1 - 2/exp(1) \\ Michel Marcus, May 01 2020

Equals Integral_{x=0..1} x/e^x dx.
Equals 1 - A135002.
Equals 1/A309419.
Equals -Integral_{x=0..1, y=0..1} x*y/(exp(x*y)*log(x*y)) dx dy. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to the above integral.) - Petros Hadjicostas, Jun 30 2020
From Amiram Eldar, Aug 05 2020: (Start)
Equals Sum_{k>=0} (-1)^k/(k! * (k+2)).
Equals Sum_{k>=1} 1/((2*k)! * (k+1)).
Equals Sum_{k>=1} (-1)^k * k^2 * H(k)/k!, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. (End)

A135004 Decimal expansion of 4/e.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 15 2007

Leading coefficient of energy difference between the two lowest states of the hydrogen molecular ion (see Holstein-Herring Method link). - Andrea Pinos, Nov 22 2023

			1.471517764685769...

G. C. Greubel, Table of n, a(n) for n = 1..2000
Shawn Godin, Problem CC35, Crux Mathematicorum, Vol. 38, No. 7 (2012), p. 265; Solution to Problem CC35, ibid., Vol. 39, No. 7 (2013), p. 299.
Wikipedia, Holstein-Herring Method.

Cf. A019741 (e/4), A068985 (1/e), A135002 (2/e).

Digits:=100: evalf(4/exp(1)); # Wesley Ivan Hurt, Jan 19 2017

RealDigits[4/E,10,120][[1]] (* Harvey P. Dale, Mar 06 2012 *)

PARI
```
4/exp(1) \\ Michel Marcus, Jan 20 2017
```

Equals lim_{n->oo} (((2*n)!/n!)^(1/n))/n (Godin, 2012). - Amiram Eldar, Apr 02 2022

More terms from Harvey P. Dale, Mar 06 2012

A227514 Decimal expansion of the square root of 2/e.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 14 2013

This appears for example while integrating the product of the absolute value of H_2(x) exp(-x^2) over the real line, where H_2 is the second Hermite polynomial.

			0.85776388496070679648018...

R. J. Mathar, Orthogonal linear combinations of Gaussian Type Orbitals, arXiv:physics/9907051 [physics.chem-ph], 1999-2009, Section VII.

Cf. A001113, A002193, A019774, A135002.

Maple
```
evalf(sqrt(2/exp(1))) ;
```

RealDigits[Sqrt[2/E], 10, 120][[1]] (* Amiram Eldar, Jul 08 2023 *)

sqrt(2/exp(1)) \\ Charles R Greathouse IV, Apr 16 2014

Square root of A135002 and also the ratio A002193 / A019774 .
From Amiram Eldar, Jul 08 2023: (Start)
Equals Product_{n>=1} (e / (1 + 1/(n-1/2))^n).
Equals Product_{n>=1} (e * (1 - 1/(n+1/2))^n). (End)

cp's OEIS Frontend

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