A139823 - OEIS

%I #7 Jun 15 2018 09:27:35

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

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

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

%N Decimal expansion of constant c = Sum_{n>=0} C(1/2^n, n).

%H G. C. Greubel, <a href="/A139823/b139823.txt">Table of n, a(n) for n = 1..5000</a>

%F c = Sum_{n>=0} log(1 + 1/2^n)^n/n! .

%e c = 1.43063452436116865706618033755902955470687309850539879176075545...

%e c = 1 + 1/2 - 3/32 + 35/1024 - 7285/524288 + 1570863/268435456 -+...

%e c = 1 + log(3/2) + log(5/4)^2/2! + log(9/8)^3/3! + log(17/16)^4/4! +...

%e The formulas for this constant illustrate the identity:

%e Sum_{n>=0} log(1 + q^n*x)^n*y^n/n! = Sum_{n>=0} binomial(q^n*y, n)*x^n.

%t RealDigits[Total[Table[Binomial[1/2^n,n],{n,0,1000}]],10,120][[1]] (* _Harvey P. Dale_, Nov 13 2014 *)

%o (PARI) a(n)=local(c=sum(m=0,n,log(1+1/2^m)^m/m!));floor(c*10^n)%10

%Y Cf. A139824, A139825.

%K cons,nonn

%O 1,2

%A _Paul D. Hanna_, May 01 2008