A269296 -id:A269296 - cp's OEIS frontend

A100734 a(n) = (5*n)!.

1, 120, 3628800, 1307674368000, 2432902008176640000, 15511210043330985984000000, 265252859812191058636308480000000, 10333147966386144929666651337523200000000

Offset: 0

Ralf Stephan, Dec 08 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..90

Cf. A000142, A010050, A100732, A100733, A268506, A269296.

[Factorial(5*n): n in [0..10]]; // Vincenzo Librandi, Sep 24 2011

(5*Range[0,10])! (* Harvey P. Dale, May 06 2015 *)

E.g.f.: 1/(1-x^5).
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ sqrt(2*Pi)*5^(5*n + 1/2)*n^(5*n + 1/2)/exp(5*n).
Sum_{n>=0} 1/a(n) = A269296. (End)

A143819 Decimal expansion of Sum_{k>=0} 1/(3*k)!.

Original entry on oeis.org

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

Offset: 1

Peter Bala, Sep 03 2008

Previous name was: Decimal expansion of the constant 1 + 1/3! + 1/6! + 1/9! + ... = 1.16805 83133 75918 ... .
Define a sequence R(n) of real numbers by R(n) := Sum_{k>=0} (3*k)^n/(3*k)! for n = 0,1,2,... . This constant is R(0); the decimal expansions of R(2) - R(1) = 1/1! + 1/4! + 1/7! and R(1) = 1/2! + 1/5! + 1/8! + ... may be found in A143820 and A143821. It is easy to verify that the sequence R(n) satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i=0..n} binomial(n,i) * 3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2) and so also an integral linear combination of R(0), R(1) and R(2) - R(1). Some examples are given below.
Bowman and Mc Laughlin (Corollary 10 with m = -1) give a continued fraction expansion for this constant and deduce the constant is irrational. - Peter Bala, Apr 17 2017

			1.168058313375918525516256929611144747716933295113292516385891232685...
R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
=======================================
  R(n)  |     R(0)     R(1)   R(2)-R(1)
=======================================
  R(3)  |       1        1        3
  R(4)  |       6        2        7
  R(5)  |      25       11       16
  R(6)  |      91       66       46
  R(7)  |     322      352      203
  R(8)  |    1232     1730     1178
  R(9)  |    5672     8233     7242
  R(10) |   32202    39987    43786
  ...
The column entries are from A143815, A143816 and A143817.

D. Bowman and J. Mc Laughlin, Polynomial continued fractions, arXiv:1812.08251 [math.NT], 2018; Acta Arith. 103 (2002), no. 4, 329-342.
Michael Penn, Two sum identities, YouTube video, 2020.
Michael I. Shamos, A catalog of the real numbers, (2011). See p. 209.

Cf. A073742, A073743, A143815, A143816, A143817, A143818, A143820, A143821.

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), this sequence (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), A269296 (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.

RealDigits[ N[ 1/3*(2*Cos[Sqrt[3]/2]/Sqrt[E] + E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)
With[{nn=120},RealDigits[N[Total[Table[1/(3n)!,{n,nn}]]+1,nn],10,nn][[1]]] (* Harvey P. Dale, Apr 20 2013 *)

suminf(k=0, 1/(3*k)!) \\ Michel Marcus, Feb 21 2016

Equals (exp(1) + exp(w) + exp(w^2))/3, where w = exp(2*Pi*i/3).
A143819 + A143820 + A143821 = exp(1).
Equals 1/3 * (e + 2 * cos(sqrt(3)/2) / sqrt(e)). - Bernard Schott, Mar 01 2020
Sum_{k>=0} (-1)^k / (3*k)! = (exp(-1) + 2*exp(1/2)*cos(sqrt(3)/2))/ 3 = 0.83471946857721... - Vaclav Kotesovec, Mar 02 2020
Continued fraction: 1 + 1/(6 - 6/(121 - 120/(505 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (3*n )*(3*n - 1)*(3*n - 2) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

Offset corrected by R. J. Mathar, Feb 05 2009

New name from Bernard Schott, Mar 02 2020

A332890 Decimal expansion of Sum_{k>=0} 1/(4*k)!.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Mar 01 2020

For q integer >= 1, Sum_{m>=0} 1/(q*m)! = (1/q) * Sum_{k=1..q} exp(X_k) where X_1, X_2, ..., X_q are the q-th roots of unity.

			1.0416914703416917479394211141000191431669197664918929...

Serge Francinou, Hervé Gianella, Serge Nicolas, Exercices de Mathématiques, Oraux X-ENS, Analyse 2, problème 3.10 p. 182, Cassini, Paris, 2004.

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 76.

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), A143819 (Sum 1/(3k)!), this sequence (Sum 1/(4k)!), A269296 (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.

Maple
```
evalf(1/2 * (cos(1) + cosh(1)), 100);
```

RealDigits[Sum[1/(4n)!,{n,0,\[Infinity]}],10,120][[1]] (* Harvey P. Dale, Apr 18 2023 *)

suminf(k=0,(1 + (-1)^k)/((2*k)!))/2 \\ Hugo Pfoertner, Mar 01 2020

suminf(k=0, 1/(4*k)!) \\ Michel Marcus, Mar 02 2020

Equals (1/2) * (cos(1) + cosh(1)).
Equals (1/2) * Sum_{k>=0} (1 + (-1)^k)/((2*k)!). - Peter Luschny, Mar 01 2020
Sum_{k>=0} (-1)^k / (4*k)! = cos(1/sqrt(2)) * cosh(1/sqrt(2)) = 0.958358132833... - Vaclav Kotesovec, Mar 02 2020
Continued fraction: 1 + 1/(24 - 24/(1681 - 1680/(11881 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (4*n)*(4*n - 1)*(4*n - 2)*(4*n - 3) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

More terms from Hugo Pfoertner, Mar 02 2020

A332892 Decimal expansion of Sum_{k>=0} 1/(6*k)!.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Mar 02 2020

For q integer >= 1, Sum_{m>=0} 1/(q*m)! = (1/q) * Sum_{k=1..q} exp(X_k) where X_1, X_2, ..., X_q are the q-th roots of unity.

			1.001388890976564743867770084409737409278656173555781142...

Serge Francinou, Hervé Gianella, Serge Nicolas, Exercices de Mathématiques, Oraux X-ENS, Analyse 2, problème 3.10, p. 182, Cassini, Paris, 2004

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), A143819 (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), A269296 (Sum 1/(5k)!), this sequence (Sum 1/(6k)!), A346441.

Maple
```
evalf(sum(1/(6*n)!,n=0..infinity),150);
```

RealDigits[(1/3)*(Cosh[1] + 2*Cosh[1/2]*Cos[Sqrt[3]/2]), 10, 120][[1]] (* Amiram Eldar, May 31 2023 *)

sumpos(k=0, 1/(6*k)!) \\ Michel Marcus, Mar 02 2020

Equals (1/3) * (cosh(1) + 2*cosh(1/2)*cos((sqrt(3))/2)).
Sum_{k>=0} (-1)^k / (6*k)! = (cos(1) + 2*cos(1/2)*cosh(sqrt(3)/2))/3 = 0.9986111131987866537... - Vaclav Kotesovec, Mar 02 2020
Continued fraction: 1 + 1/(720 - 720/(665281 - 665280/(13366081 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (6*n)*(6*n - 1)*(6*n - 2)*(6*n - 3)*(6*n - 4)*(6*n - 5) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

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A100734 a(n) = (5*n)!.

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A143819 Decimal expansion of Sum_{k>=0} 1/(3*k)!.

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A332890 Decimal expansion of Sum_{k>=0} 1/(4*k)!.

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A332892 Decimal expansion of Sum_{k>=0} 1/(6*k)!.

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Maple

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