Sum of reciprocals of double factorials

[Eric W. Weisstein]

On Wed, 6 Aug 2008, Jonathan Post wrote:

> Perhaps it is now worth submitting the two sequences of numerators and
> denominators of the convergents to this real number, given that Eric
> Wesstien has added to MathWorld and OEIS with Max Alekseyev's lovely
> formula:
>
> Weisstein, Eric W. "Double Factorial." From MathWorld--A Wolfram Web
> Resource. http://mathworld.wolfram.com/DoubleFactorial.html
>
> "The sum of reciprocal double factorials can be given in closed form as
> sum_(n=0)^infty1/(n!!)=sqrt(e)[1+sqrt(pi/2)erf(1/2sqrt(2))]"
> and
>
> (Sloane's A143280).

Turns out the development version of Mathematica gets that one out of the 
box.

> This leaves open the analogous:
>
> sum_(n=0)^infty1/(n!!!)

That's a bit trickier.  Mathematica doesn't get it quite out of the box, 
but with some human help, the result is

1/3 E^(1/3) (3 + 3^(2/3) Gamma[2/3] + 3 3^(1/3) Gamma[4/3] -
    3^(1/3) Gamma[1/3, 1/3] - 3^(2/3) Gamma[2/3, 1/3])

(probably can be simplified somewhat)

== 3.2989135380884190034012178082614697690778036956832...

> sum_(n=0)^infty1/(n!!!!)
>
> sum_(n=0)^infty1/(n!!!!!)
>
> etcetera, and their convergents.

Ah; what's very nice about the above formula for the n == 3 case is that 
you can immediately see the pattern for the general case:

ReciprocalFactorialSumConstant[n_] :=
  1/n Exp[1/n] (n + Sum[n^(k/n) Gamma[k/n, 0, 1/n], {k, n - 1}])

In[2]:= ReciprocalFactorialSumConstant[2] // FullSimplify
Out[2]= Sqrt[E] + Sqrt[(E \[Pi])/2] Erf[1/Sqrt[2]]

In[3]:= ReciprocalFactorialSumConstant[3]
Out[3]= 1/3 E^(1/3) (3 + 3^(1/3) Gamma[1/3, 0, 1/3] +
    3^(2/3) Gamma[2/3, 0, 1/3])

In[4]:= N[%, 20]
Out[4]= 3.2989135380884190034

In[5]:= ReciprocalFactorialSumConstant[4]
1/4 E^(1/4) (4 + Sqrt[2] Gamma[1/4, 0, 1/4] + 2 Gamma[1/2, 0, 1/4] +
    2 Sqrt[2] Gamma[3/4, 0, 1/4])

In[6]:= N[%, 20]
Out[6]= 3.4859449774535577452

etc.

Cheers,
-Eric

> On Mon, Aug 4, 2008 at 12:41 PM, Jonathan Post <jvospost3 at gmail.com> wrote:
>> Very elegant, Max Alekseyev!
>>
>> The generalization to k-tuple factorials is similar, and the
>> convergents require a modulo k adjustment?
>>
>> In my earlier email, of course I meant "What are the real numbers to
>> which the sum of reciprocals of k-tuple
>> factorials converges?"
>>
>> With your formulae, is this now worth submitting, as definitive rather
>> than merely an analogue of existing seqs?