A246900 Decimal expansion of the constant c = Sum_{n>=0} binomial(n-1 + 1/2^(n-1), n).

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

Offset

1,1

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1001

Formula

c = Sum_{n>=0} (-2)^n * log(1 - 1/2^n)^n / n!.

c = Sum_{n>=0} A224883(n) / 2^(n^2), where A224883(n) = (2^n/n!) * Product_{k=0..n-1} (2^(n-1)*k + 1).

Example

c = 2.55500248436101360804704969796239525102504151483916927730...
where the constant is equal to the sum
c = 1 + binomial(1,1) + binomial(3/2,2) + binomial(9/4,3) + binomial(25/8,4) + binomial(65/16,5) + binomial(161/32,6) +...+ binomial(n-1 + 1/2^(n-1), n) +...
which may be written as
c = 1 + 2/2 + 6/2^4 + 60/2^9 + 2550/2^16 + 476476/2^25 + 384115732/2^36 + 1305385229720/2^49 + 18382187112952806/2^64 +...+ A224883(n)*x^n/2^(n^2) +...
The constant also equals the logarithmic sum
c = 1 + 2*log(2) + 4*log(4/3)^2/2! + 8*log(8/7)^3/3! + 16*log(16/15)^4/4! + 32*log(32/31)^5/5! + 64*log(64/63)^6/6! +...+ (-2)^n*log(1 - 1/2^n)^n/n! +...
which converges rather quickly.

Prog

(PARI) /* By definition: */
\p128
{c=suminf(n=0, binomial(n-1 + 1/2^(n-1), n)*1.)}
{a(n)=floor(10^n*c)%10}
for(n=0, 120, print1(a(n), ", "))
(PARI) /* By a logarithmic identity (accelerated series): */
\p1024
{c=1+suminf(n=1, (-2)^n*log(1 - 1/2^n)^n / n!)}
{a(n)=floor(10^n*c)%10}
for(n=0, 1000, print1(a(n), ", "))

Crossrefs

Cf. A224883.

Sequence in context: A197004 A363764 A116698 * A277086 A229710 A240947

Adjacent sequences: A246897 A246898 A246899 * A246901 A246902 A246903

Keyword

nonn,cons

Author

Paul D. Hanna, Nov 29 2014

Status

approved