cp's OEIS Frontend

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

%I A246900 #48 Nov 29 2014 21:09:43
%S A246900 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,
%T A246900 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,
%U A246900 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
%N A246900 Decimal expansion of the constant c = Sum_{n>=0} binomial(n-1 + 1/2^(n-1), n).
%H A246900 Paul D. Hanna, <a href="/A246900/b246900.txt">Table of n, a(n) for n = 1..1001</a>
%F A246900 c = Sum_{n>=0} (-2)^n * log(1 - 1/2^n)^n / n!.
%F A246900 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).
%e A246900 c = 2.55500248436101360804704969796239525102504151483916927730...
%e A246900 where the constant is equal to the sum
%e A246900 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) +...
%e A246900 which may be written as
%e A246900 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) +...
%e A246900 The constant also equals the logarithmic sum
%e A246900 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! +...
%e A246900 which converges rather quickly.
%o A246900 (PARI) /* By definition: */
%o A246900 \p128
%o A246900 {c=suminf(n=0,binomial(n-1 + 1/2^(n-1), n)*1.)}
%o A246900 {a(n)=floor(10^n*c)%10}
%o A246900 for(n=0,120,print1(a(n),", "))
%o A246900 (PARI) /* By a logarithmic identity (accelerated series): */
%o A246900 \p1024
%o A246900 {c=1+suminf(n=1, (-2)^n*log(1 - 1/2^n)^n / n!)}
%o A246900 {a(n)=floor(10^n*c)%10}
%o A246900 for(n=0,1000,print1(a(n),", "))
%Y A246900 Cf. A224883.
%K A246900 nonn,cons
%O A246900 1,1
%A A246900 _Paul D. Hanna_, Nov 29 2014