A139823 Decimal expansion of constant c = Sum_{n>=0} C(1/2^n, n).
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, 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, 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
Offset: 1
Examples
c = 1.43063452436116865706618033755902955470687309850539879176075545...
c = 1 + 1/2 - 3/32 + 35/1024 - 7285/524288 + 1570863/268435456 -+...
c = 1 + log(3/2) + log(5/4)^2/2! + log(9/8)^3/3! + log(17/16)^4/4! +...
The formulas for this constant illustrate the identity:
Sum_{n>=0} log(1 + q^n*x)^n*y^n/n! = Sum_{n>=0} binomial(q^n*y, n)*x^n.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..5000
Programs
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Mathematica
RealDigits[Total[Table[Binomial[1/2^n,n],{n,0,1000}]],10,120][[1]] (* Harvey P. Dale, Nov 13 2014 *) -
PARI
a(n)=local(c=sum(m=0,n,log(1+1/2^m)^m/m!));floor(c*10^n)%10
Formula
c = Sum_{n>=0} log(1 + 1/2^n)^n/n! .