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Reciprocal Multifactorial Constant

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The reciprocal multifactorial constant of positive integer order n is the sum of reciprocal multifactorials of order n and has closed form

m(n)=sum_(k=0)^(infty)1/(k!...!_()_(n))
(1)
=1+(e^(1/n))/nsum_(k=1)^(n)n^(k/n)gamma(k/n,1/n)
(2)
=(e^(1/n))/n[n+sum_(k=1)^(n-1)n^(k/n)gamma(k/n,1/n)],
(3)

where gamma(a,x) is a lower incomplete gamma function (E. W. Weisstein, Aug. 6, 2008).

For n=2 and 3, the special cases are given by OEIS A143280 and OEIS A288055, respectively,

m(2)=sqrt(e)[1+sqrt(pi/2)erf(1/2sqrt(2))]
(4)
m(3)=1/3e^(1/3)[3+3^(1/3)gamma(1/3,1/3)+3^(2/3)gamma(2/3,1/3)].
(5)

See also

Multifactorial

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References

Sloane, N. J. A. Sequences A143280 and A288055in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Reciprocal Multifactorial Constant

Cite this as:

Weisstein, Eric W. "Reciprocal Multifactorial Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ReciprocalMultifactorialConstant.html

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