%I #49 Nov 06 2024 13:33:03

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

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

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

%N Decimal expansion of m(2) = Sum_{n>=0} 1/n!!.

%C Also decimal expansion of Sum_{n>=1} n!!/n!. - _Michel Lagneau_, Dec 24 2011

%C Apart from the first digit, the same as A227569. - _Robert G. Wilson v_, Apr 09 2014

%H G. C. Greubel, <a href="/A143280/b143280.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael Penn, <a href="https://www.youtube.com/watch?v=LGsXnm5sYow">Finding the closed form for a double factorial sum</a>, YouTube video, 2022.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ReciprocalMultifactorialConstant.html">Reciprocal Multifactorial Constant</a>

%F Equals sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)).

%e 3.05940740534257614453947549923327861297725472633534020929971877980544281968...

%t RealDigits[ Sqrt[E] + Sqrt[E*Pi/2]*Erf[1/Sqrt[2]], 10, 105][[1]] (* or *)

%t RealDigits[ Sum[1/n!!, {n, 0, 125}], 10, 105][[1]] (* _Robert G. Wilson v_, Apr 09 2014 *)

%t RealDigits[Total[1/Range[0,200]!!],10,120][[1]] (* _Harvey P. Dale_, Apr 10 2022 *)

%o (PARI) default(realprecision, 100); exp(1/2)*(1 + sqrt(Pi/2)*(1-erfc(1/sqrt(2) ))) \\ _G. C. Greubel_, Mar 27 2019

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Exp(1/2)*(1 + Sqrt(Pi(R)/2)*Erf(1/Sqrt(2) )); // _G. C. Greubel_, Mar 27 2019

%o (Sage) numerical_approx(exp(1/2)*(1 + sqrt(pi/2)*erf(1/sqrt(2))), digits=100) # _G. C. Greubel_, Mar 27 2019

%Y Cf. A227569.

%Y Cf. A006882 (n!!), this sequence (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

%K nonn,cons,changed

%O 1,1

%A _Eric W. Weisstein_, Aug 04 2008