A143280 -id:A143280 - cp's OEIS frontend

A334367 Decimal expansion of Sum_{k>=0} 1/(4*k+2)!!.

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

Offset: 0

Ilya Gutkovskiy, Apr 24 2020

This constant is transcendental.

			1/(2^1*1!) + 1/(2^3*3!) + 1/(2^5*5!) + ... = 0.52109530549374736162242...

Index entries for transcendental numbers

Cf. A019774, A067626, A073742, A143280, A160327, A201504, A334366.

Mathematica
```
RealDigits[Sinh[1/2], 10, 110] [[1]]
```
PARI
```
sinh(1/2) \\ Michel Marcus, Apr 25 2020
```

Equals sinh(1/2).

Equals (1/2) * Product_{k>=1} 1 + 1/(2*k*Pi)^2. - Amiram Eldar, Jul 16 2020

A227569 Decimal expansion of maximal value of function F[a(n); b(n)] for pairs of complements a(n) and b(n) of natural numbers A000027, where a(n) = odd numbers (A005408) and b(n) = even numbers (A005843); see Comments for the definition of function F[a(n); b(n)].

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Jul 16 2013

Apart from the first digit, the same as A143280. The sum of the reciprocals of the double factorial numbers, Sum_{n>=1} 1/n!! = Sum_{n>=2} n!!/n!. - Robert G. Wilson v, Jun 27 2015
Definition of function F[a(n); b(n)]: Let a(n) and b(n) is pair of complements of natural numbers (A000027) with a(1) < a(2) < a(3) < ... and b(1) < b(2) < b(3) < ..., then F[a(n); b(n)] = F[a(n)] + F[b(n)]; where F[a(n)] = 1/a(1) + 1/a(1)a(2) + 1/a(1)a(2)a(3) + ... and F[b(n)] = 1/b(1) + 1/b(1)b(2) + 1/b(1)b(2)b(3) + ...
Value of function F[a(n); b(n)] is real number c = a + b, where a = real number whose Engel expansion is sequence a(n) and b = real number whose Engel expansion is sequence b(n). See A006784 for definition of Engel expansion.
Example for a(n) = odd numbers (A005408) and b(n) = even numbers (A005843): c = 2.059407... = a + b, where a = 1.410686... (A060196) and b = 0.648721... (A019774 - 1).
Example for a(n) = nonprime numbers (A018252) and b(n) = primes (A000040): c = 2.002747... = a + b, where a = 1.297516... and b = 0.705230... (A064648).
Conjecture: there are no pairs of complements a(n) and b(n) such that F[a(n); b(n)] = 2.
e - 1 <= F[a(n); b(n)] <= sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)) - 1.
1.71828182... (A091131) <= F[a(n); b(n)] <= 2.05940740....

			2.05940740534257614453947549923327861297725472633534020929971877980544281968...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Engel Expansion
Googology Wiki, Double factorial

Cf. A000027, A005408, A005843, A091131 (e-1), A006882 (n!!), A143280 (m(2)).

SetDefaultRealField(RealField(112)); R:= RealField(); -1 + Exp(1/2)*(1 + Sqrt(Pi(R)/2)*Erf(1/Sqrt(2)) ); // G. C. Greubel, Apr 01 2019

RealDigits[Sqrt[E] -1 + Sqrt[E*Pi/2]*Erf[1/Sqrt[2]], 10, 105][[1]] (* or *)
RealDigits[Sum[1/n!!, {n, 125}], 10, 105][[1]] (* Robert G. Wilson v, Apr 09 2014 *)

default(realprecision, 100); exp(1/2) - 1 + sqrt(exp(1)*Pi/2)*(1-erfc(1/sqrt(2))) \\ G. C. Greubel, Apr 01 2019

numerical_approx(-1 + exp(1/2)*(1 + sqrt(pi/2)*erf(1/sqrt(2))), digits=112) # G. C. Greubel, Apr 01 2019

A202688 Decimal expansion of Sum_{n>=0} (-1)^n / n!!.

Original entry on oeis.org

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

Offset: 0

Michel Lagneau, Dec 24 2011

			0.23803513605768014915782607639504...

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

Cf. A006882 (n!!), A143280 (m(2)).

SetDefaultRealField(RealField(112)); R:= RealField(); Exp(1/2)*(1 - Sqrt(Pi(R)/2)*Erf(1/Sqrt(2)) ); // G. C. Greubel, Mar 28 2019

with(numtheory):Digits:=200:s:=evalf(sum(‘((-1)^(i+1))*doublefactorial(i)/i! ’,’i’=1..100)):print(s):

RealDigits[N[Sum[((-1)^(n+1))/n!!,{n,0,100}],105]][[1]]
RealDigits[Sqrt[E] - Sqrt[(E*Pi)/2]*Erf[1/Sqrt[2]], 10, 105][[1]] (* G. C. Greubel, Mar 28 2019 *)

exp(.5) - sqrt(exp(1)*Pi/2)*(1-erfc(sqrt(.5))) \\ Charles R Greathouse IV, Nov 21 2016

numerical_approx(exp(1/2)*(1 - sqrt(pi/2)*erf(1/sqrt(2))), digits=112) # G. C. Greubel, Mar 28 2019

Equals Sum_{n>=1} (-1)^(n+1)*n!! /n!.

Equals sqrt(e) - sqrt(e*Pi/2)*erf(1/sqrt(2)).

Terms a(80) onward corrected by G. C. Greubel, Mar 28 2019

Name corrected by Thomas Ordowski, Oct 22 2024

A334366 Decimal expansion of Sum_{k>=0} 1/(4*k)!!.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 24 2020

This constant is transcendental.

			1/(2^0*0!) + 1/(2^2*2!) + 1/(2^4*4!) + ... = 1.1276259652063807852...

Index entries for transcendental numbers

Cf. A019774, A067624, A073743, A143280, A160327, A201505, A334367.

Mathematica
```
RealDigits[Cosh[1/2], 10, 110] [[1]]
```
PARI
```
cosh(1/2) \\ Michel Marcus, Apr 25 2020
```

Equals cosh(1/2).

Equals Product_{k>=0} 1 + 1/((2*k+1)*Pi)^2. - Amiram Eldar, Jul 16 2020

A342033 Decimal expansion of m(10) = Sum_{n>=0} 1/n!10, the 10th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Bhoris Dhanjal, Feb 26 2021

m(k) can be proved to approach a harmonic series (and diverge) as k approaches infinity.

			4.165243765558384590787262...
For n=10, the series is equal to 1+summation from n=1 to 10 (1/n)=9901/2520.

Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

Multifactorial[n_, k_] := Abs[Apply[Times, Range[-n, -1, k]]]
N[Sum[1/Multifactorial[n, 10], {n, 0, 10000}], 105]
(* or *)
ReciprocalFactorialSumConstant[k_] :=
1/k Exp[1/k] (k + Sum[k^(j/k) Gamma[j/k, 0, 1/k], {j, k - 1}])
N[ReciprocalFactorialSumConstant[10], 105]

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} (k^(j/k)*Gamma(j/k, 1/k))) where Gamma(a,x) the incomplete Gamma function.

A358361 Decimal expansion of the constant Sum_{j>=0} j!!/(2*j)!, where j!! indicates the double factorial of j.

Original entry on oeis.org

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

Offset: 1

Marco Ripà, Nov 12 2022

Sum_{j>=0} j!!/(2j)! converges since Sum_{j>=0} j!!/j! converges by A143280 (and it is trivial to note that (2*j)! >= j! for any positive integer j).

			1.587702647727...

Eric Weisstein's World of Mathematics, Double Factorial.

Cf. A006882, A010050, A143280, A264152.

RealDigits[1 + Sum[(i)!!/(2i)!, {i,1,200}], 10, 105][[1]]

Equals Sum_{j>=0} A006882(j)/A010050(j).

Equals Sum_{j>=0} 1/(A264152(j)*j!).

cp's OEIS Frontend

A334367 Decimal expansion of Sum_{k>=0} 1/(4*k+2)!!.

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A227569 Decimal expansion of maximal value of function F[a(n); b(n)] for pairs of complements a(n) and b(n) of natural numbers A000027, where a(n) = odd numbers (A005408) and b(n) = even numbers (A005843); see Comments for the definition of function F[a(n); b(n)].

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A202688 Decimal expansion of Sum_{n>=0} (-1)^n / n!!.

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A334366 Decimal expansion of Sum_{k>=0} 1/(4*k)!!.

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A342033 Decimal expansion of m(10) = Sum_{n>=0} 1/n!10, the 10th reciprocal multifactorial constant.

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Mathematica

Formula

A358361 Decimal expansion of the constant Sum_{j>=0} j!!/(2*j)!, where j!! indicates the double factorial of j.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula