A332890 -id:A332890 - cp's OEIS frontend

A100733 a(n) = (4*n)!.

1, 24, 40320, 479001600, 20922789888000, 2432902008176640000, 620448401733239439360000, 304888344611713860501504000000, 263130836933693530167218012160000000, 371993326789901217467999448150835200000000, 815915283247897734345611269596115894272000000000

Offset: 0

Ralf Stephan, Dec 08 2004

From Karol A. Penson, Jun 11 2009: (Start)
Integral representation of a(n) as n-th moment of a positive function W(x) = (1/4)*exp(-x^(1/4))/x^(3/4) on the positive axis:
a(n) = Integral_{x=0..oo} x^n*W(x) dx = Integral_{x=0..oo} x^n*(1/4)*exp(-x^(1/4))/x^(3/4) dx, n >= 0.
This is the solution of the Stieltjes moment problem with the moments a(n), n >= 0.
As the moments a(n) grow very rapidly this suggests, but does not prove, that this solution may not be unique.
This is indeed the case as by construction the following "doubly" infinite family:
V(k,a,x) = (1/4)*exp(-x^(1/4))*(a*sin((3/4)*Pi*k + tan((1/4)*Pi*k)*x^(1/4)) + 1)/x^(3/4),
with the restrictions k=+-1,+-2,..., abs(a) < 1 is still positive on 0 <= x < infinity and has moments a(n).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A000142, A010050, A100732, A100734, A268505, A332890.

[Factorial(4*n): n in [0..10]]; // Vincenzo Librandi, Sep 24 2011

(4*Range[0,20])! (* Harvey P. Dale, Oct 03 2014 *)

E.g.f.: 1/(1-x^4).
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ sqrt(Pi)*2^(8*n+3/2)*n^(4*n+1/2)/exp(4*n).
Sum_{n>=0} 1/a(n) = (cos(1) + cosh(1))/2 = 1.04169147034169174... = A332890. (End)
Sum_{n>=0} (-1)^n/a(n) = cos(1/sqrt(2))*cosh(1/sqrt(2)). - Amiram Eldar, Feb 14 2021

More terms from Harvey P. Dale, Oct 03 2014

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

A337727 a(n) = (4n)! Sum_{k=0..n} 1 / (4*k)!.

Original entry on oeis.org

1, 25, 42001, 498971881, 21795091762081, 2534333270094778681, 646315807872650838343345, 317599587988620621961919733001, 274101148417699141578015206369183041, 387502275541069630431671657548241448722521, 849931991080760484603611346800010863970028660561

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A100733, A330045, A332890, A337725, A337726, A337728, A337729, A337730.

Table[(4 n)! Sum[1/(4 k)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n)! SeriesCoefficient[(1/2) (Cos[x] + Cosh[x])/(1 - x^4), {x, 0, 4 n}], {n, 0, 10}]
Table[Floor[(1/2) (Cos[1] + Cosh[1]) (4 n)!], {n, 0, 10}]

a(n) = (4*n)!*sum(k=0, n, 1/(4*k)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (cos(x) + cosh(x)) / (1 - x^4) = 1 + 25*x^4/4! + 42001*x^8/8! + 498971881*x^12/12! + ...

a(n) = floor(c * (4*n)!), where c = (cos(1) + cosh(1)) / 2 = A332890.

A269296 Decimal expansion of Sum_{k>=0} 1/(5k)!.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Feb 21 2016

From Vaclav Kotesovec, Feb 24 2016: (Start)
Sum_{k>=0} 1/k! = A001113 = exp(1).
Sum_{k>=0} 1/(2k)! = A073743 = cosh(1).
Sum_{k>=0} 1/(3k)! = A143819 = (2*cos(sqrt(3)/2)*exp(-1/2) + exp(1))/3.
Sum_{k>=0} 1/(4k)! = (cos(1) + cosh(1))/2 = 1.0416914703416917479394211141...
(End)
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. - Bernard Schott, Mar 02 2020
Continued fraction: 1 + 1/(120 - 120/(30241 - 30240/(360361 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (5*n)*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

			1 + 1/5! + 1/10! + 1/15! + ... = 1.008333608907290289976453667354838786...

Eric Weisstein's World of Mathematics, Factorial Sums

Cf. A100734.

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), A143819 (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), this sequence (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.

evalf((exp(1) + 2*exp(-(sqrt(5) + 1)/4) * cos(sqrt((5 - sqrt(5))/2)/2) + 2*exp((sqrt(5) - 1)/4) * cos(sqrt((5 + sqrt(5))/2)/2))/5, 120); # Vaclav Kotesovec, Feb 24 2016

RealDigits[HypergeometricPFQ[{}, {1/5, 2/5, 3/5, 4/5}, 1/3125], 10, 120][[1]]

suminf(k=0, 1/(5*k)!) \\ Michel Marcus, Feb 21 2016

Equals Sum_{k>=0} 1/A100734(k).
Equals (exp(1) + exp(-(-1)^(1/5)) + exp((-1)^(2/5)) + exp(-(-1)^(3/5)) + exp((-1)^(4/5)))/5.
Equals (exp(1) + 2*exp(-(sqrt(5) + 1)/4) * cos(sqrt((5 - sqrt(5))/2)/2) + 2*exp((sqrt(5) - 1)/4) * cos(sqrt((5 + sqrt(5))/2)/2))/5. - Vaclav Kotesovec, Feb 24 2016
Sum_{k>=0} (-1)^k / (5*k)! = (exp(-1) + 2*cos(5^(1/4)/(2*sqrt(phi))) * exp(phi/2) + 2*cos(5^(1/4)*sqrt(phi)/2) / exp(1/(2*phi)))/5 = 0.99166694223909419..., where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, 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

A334363 Decimal expansion of Sum_{k>=0} 1/(4*k+1)!.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 24 2020

			1/1! + 1/5! + 1/9! + ... = 1.008336089225848981767442...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 36.

Cf. A001113, A049469, A073742, A143820, A332890, A334364, A334365.

RealDigits[(Sin[1] + Sinh[1])/2, 10, 110] [[1]]

suminf(k=0, 1/(4*k+1)!) \\ Michel Marcus, Apr 25 2020

Equals (sin(1) + sinh(1))/2.

A334364 Decimal expansion of Sum_{k>=0} 1/(4*k+2)!.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 24 2020

			1/2! + 1/6! + 1/10! + ... = 0.5013891644735520305...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 457.

Cf. A001113, A049470, A073743, A143821, A332890, A334363, A334365.

RealDigits[(Cosh[1] - Cos[1])/2, 10, 110] [[1]]

suminf(k=0, 1/(4*k+2)!) \\ Michel Marcus, Apr 25 2020

Equals (cosh(1) - cos(1))/2.

A334365 Decimal expansion of Sum_{k>=0} 1/(4*k+3)!.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 24 2020

			1/3! + 1/7! + 1/11! + ... = 0.1668651044179524751...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 209.
Index entries for transcendental numbers.

Cf. A001113, A049469, A073742, A332890, A334363, A334364.

RealDigits[(Sinh[1] - Sin[1])/2, 10, 111] [[1]]

suminf(k=0, 1/(4*k+3)!) \\ Michel Marcus, Apr 25 2020

Equals (sinh(1) - sin(1))/2.

A352660 a(n) = n! * Sum_{k=0..floor(n/4)} 1 / (4*k)!.

Original entry on oeis.org

1, 1, 2, 6, 25, 125, 750, 5250, 42001, 378009, 3780090, 41580990, 498971881, 6486634453, 90812882342, 1362193235130, 21795091762081, 370516559955377, 6669298079196786, 126716663504738934, 2534333270094778681, 53220998671990352301, 1170861970783787750622

Offset: 0

Ilya Gutkovskiy, Mar 25 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000522, A009179, A332890, A337727, A349087, A352659.

Table[n! Sum[1/(4 k)!, {k, 0, Floor[n/4]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(Cos[x] + Cosh[x])/(2 (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = n! * sum(k=0, n\4, 1/(4*k)!); \\ Michel Marcus, Mar 29 2022

E.g.f.: (cos(x) + cosh(x)) / (2*(1 - x)).

a(n) = floor(c * n!), where c = 1.04169147... = A332890.

cp's OEIS Frontend

A100733 a(n) = (4*n)!.

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

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A337727 a(n) = (4*n)! * Sum_{k=0..n} 1 / (4*k)!.

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A269296 Decimal expansion of Sum_{k>=0} 1/(5k)!.

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

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A334363 Decimal expansion of Sum_{k>=0} 1/(4*k+1)!.

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A334364 Decimal expansion of Sum_{k>=0} 1/(4*k+2)!.

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A337727 a(n) = (4n)! Sum_{k=0..n} 1 / (4*k)!.