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 A246900 Decimal expansion of the constant c = Sum_{n>=0} binomial(n-1 + 1/2^(n-1), n). 2
 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 (list; constant; graph; refs; listen; history; text; internal format)
 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: A200484 A197004 A116698 * A277086 A229710 A240947 Adjacent sequences:  A246897 A246898 A246899 * A246901 A246902 A246903 KEYWORD nonn,cons AUTHOR Paul D. Hanna, Nov 29 2014 STATUS approved

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Last modified June 15 22:06 EDT 2021. Contains 345053 sequences. (Running on oeis4.)