Reciprocal Multifactorial Constant

Author

Eric W. Weisstein
August 13, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/ReciprocalMultifactorialConstant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ReciprocalMultifactorialConstant.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Code

Multifactorial[n_,k_]:=Times@@Range[n,1,-k]

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

ReciprocalFactorialSumConstant2[n_]:=(1/n)Exp[1/n]Sum[(n^(k/n))Gamma[(k/n),0,(1/n)],{k,n}]+1

Factorial

Sum[(1/(n!)),{n,0,∞}]

Double Factorial

Sum[(1/(Multifactorial[n,2])),{n,0,20}]//N

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

Triple Factorial

Sum[(1/(Multifactorial[n,3])),{n,0,20}]//N

Factorial3[n_]:=Piecewise[{
{1,n==0},
{(((-3))^((1/3) (-3+n))) n Pochhammer[1-(n/3),(1/3) (-3+n)],Mod[n,3]==0},
{(((-3))^((1/3) (-1+n))) n Pochhammer[1-(n/3),(1/3) (-1+n)],Mod[n,3]==1},
{(((-3))^((1/3) (-2+n))) n Pochhammer[1-(n/3),(1/3) (-2+n)],Mod[n,3]==2}}]

Sum[(1/(Factorial3[n])),{n,∞}]

ReciprocalFactorialSumConstant[3]

N[%,50]

N[Sum[(1/(Factorial3[n])),{n,0,100}],50]

N[Sum[(1/(Factorial3[n])),{n,0,200}],50]

≡ 0 (mod 3)

FullSimplify[(((-3))^((1/3) (-3+n))) n Pochhammer[1-(n/3),(1/3) (-3+n)],Mod[n,3]==0&&n>0]

Table[(((-3))^((1/3) (-3+n))) n Pochhammer[1-(n/3),-1+(n/3)],{n,3,20,3}]

FullSimplify[(((-3))^((1/3) (-3+n))) n Pochhammer[1-(n/3),(1/3) (-3+n)]/.n->3n,n∈Integers&&n>0]/.n->n/3

Table[(3^(n/3)) (n/3)!,{n,3,20,3}]

Sum[(1/((((-3))^((1/3) (-3+n))) n Pochhammer[1-(n/3),(1/3) (-3+n)])),{n,∞}]

Table[(1/((((-3))^((1/3) (-3+n))) n Pochhammer[1-(n/3),(1/3) (-3+n)])),{n,20}]

N[Sum[(1/((((-3))^((1/3) (-3+n))) n Pochhammer[1-(n/3),(1/3) (-3+n)])),{n,10^3}],20]

Simplify[(1/((((-3))^((1/3) (-3+n))) n Pochhammer[1-(n/3),(1/3) (-3+n)]))/.n->3n]

FullSimplify[(1/((((-3))^((1/3) (-3+n))) n Pochhammer[1-(n/3),(1/3) (-3+n)]))/.n->3n,n∈Integers&&n>0]

Table[((3^(-n))/(n!)),{n,10}]

s0=Sum[((3^(-n))/(n!)),{n,∞}]

N[%]

≡ 1 (mod 3)

Table[(((-3))^((1/3) (-1+n))) n Pochhammer[1-(n/3),(1/3) (-1+n)],{n,1,20,3}]

FullSimplify[(((-3))^((1/3) (-1+n))) n Pochhammer[1-(n/3),(1/3) (-1+n)]/.n->3k+1,k∈Integers&&k>0]

Table[(((-3))^k) (1+3 k) Pochhammer[(2/3)-k,k],{k,5}]

s1=Sum[(1/((((-3))^k) (1+3 k) Pochhammer[(2/3)-k,k])),{k,0,∞}]

≡ 2 (mod 3)

Table[(((-3))^((1/3) (-2+n))) n Pochhammer[1-(n/3),(1/3) (-2+n)],{n,2,20,3}]

FullSimplify[(((-3))^((1/3) (-2+n))) n Pochhammer[1-(n/3),(1/3) (-2+n)]/.n->3k+2,k∈Integers&&k>0]

Table[(((-3))^k) (2+3 k) Pochhammer[(1/3)-k,k],{k,0,5}]

s2=Sum[(1/((((-3))^k) (2+3 k) Pochhammer[(1/3)-k,k])),{k,0,∞}]

All

Factorial3[0]

sum=1+s0+s1+s2

N[sum,50]

GammaSimplify[Gamma[2/3]]

FullSimplify[sum]

Simplify[Expand[sum]]

N[(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)]),50]

SimplifyCount[p_]:=
Which[Head[p]===Symbol,1,
IntegerQ[p],If[p==0,1,Floor[N[Log[2,Abs[p]]/Log[2,10]]]+If[p>0,1,2]],
Head[p]===Rational,SimplifyCount[Numerator[p]]+SimplifyCount[Denominator[p]]+1,
Head[p]===Complex,SimplifyCount[Re[p]]+SimplifyCount[Im[p]]+1,NumberQ[p],2,
True,SimplifyCount[Head[p]]+If[Length[p]==0,0,Plus@@(SimplifyCount/@(List@@p))]]

f[e_]:=100Count[e,_ExpIntegralE,{0,Infinity}]+SimplifyCount[e]

FullSimplify[%278,ComplexityFunction->f]

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

n!!!!

N[Sum[(1/(Multifactorial[n,4])),{n,0,300}],50]

N[(1/4) (E^(1/4)) (4+(4^(1/4)) Gamma[(1/4)]+(4^(2/4)) Gamma[(2/4)]+(4^(3/4))Gamma[(3/4)]-(4^(1/4)) Gamma[(1/4),(1/4)]-(4^(2/4)) Gamma[(2/4),(1/4)]-(4^(3/4)) Gamma[(3/4),(1/4)]),50]

General

Sum[(n^(k/n))Gamma[(k/n),0,(1/n)],{k,n}]

Sum[(n^(k/n))Gamma[(k/n),0,(1/n)],{k,n-1}]

Gamma[(k/n),0,(1/n)]/.k->n

ReciprocalFactorialSumConstant[2]

ReciprocalFactorialSumConstant[2]//FullSimplify

ReciprocalFactorialSumConstant[2]//FunctionExpand//FullSimplify

N[%,20]

ReciprocalFactorialSumConstant2[3]

ReciprocalFactorialSumConstant[3]

FullSimplify[FunctionExpand[%]]

N[%,20]

ReciprocalFactorialSumConstant[4]

FullSimplify[%]

FullSimplify[FunctionExpand[%]]

N[%,20]

l=Table[ReciprocalFactorialSumConstant[n],{n,500}];//Timing

ListPlot[l,Joined->True,PlotRange->All]

SequenceLimit[N[l,100]]