0,2

a(n) = [x^n] {(1-sqrt(1 - 2^(n+3)*x))/(2^(n+2)*x)}^(1/2^n).

G.f.: A(x) = Sum_{n>=0} a(n)*x^n/2^(n^2+n).

G.f.: A(x) = Sum_{n>=0} log(F(x/2^n))^n/n! where F(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).

Radius of convergence of A(x) is |x| <= 1/2.

G.f.: A(x) = 1 + 2*x/2^2 + 26*x^2/2^6 + 1804*x^3/2^12 + 591894*x^4/2^20 +...

G.f.: A(x) = Sum_{n>=0} log( 2^n*(1-sqrt(1 - 4*x/2^n))/(2*x) )^n/n!.

A(x) = 1 + log(F(x/2)) + log(F(x/4))^2/2! + log(F(x/8))^3/3! +... where F(x) = (1-sqrt(1 - 4*x))/(2*x).

Special values.

A(1/2) = 1 + log(2) + log(4-4*sqrt(1/2))^2/2! + log(8-8*sqrt(3/4))^3/3! + log(16-16*sqrt(7/8))^4/4! +...

A(1/2) = 1.70573970062357248928512380703308976974285275...

A(-1/2) = 1 + log(2*sqrt(2)-2) + log(4*sqrt(3/2)-4)^2/2! + log(8*sqrt(5/4)-8)^3/3! + log(16*sqrt(9/8)-16)^4/4! +...

A(-1/2) = 0.81741280310249092844743171863299249334671633...

Illustrate a(n) = [x^n] {(1-sqrt(1-2^(n+3)*x))/(2^(n+2)*x)}^(1/2^n):

n=0: (1) + 2*x + 8*x^2 + 40*x^3 + 224*x^4 + 1344*x^5 +...

n=1: 1 + (2)*x + 14*x^2 + 132*x^3 + 1430*x^4 + 16796*x^5 +...

n=2: 1 + 2*x + (26)*x^2 + 476*x^3 + 10150*x^4 + 236060*x^5 +...

n=3: 1 + 2*x + 50*x^2 + (1804)*x^3 + 76342*x^4 + 3534076*x^5 +...

n=4: 1 + 2*x + 98*x^2 + 7020*x^3 + (591894)*x^4 + 54673468*x^5 +...

n=5: 1 + 2*x + 194*x^2 + 27692*x^3 + 4660950*x^4 + (860081340)*x^5 +...

coefficients in parenthesis form the initial terms of this sequence.

(PARI) a(n)=2^(n^2+n)*binomial(2*n-1+1/2^n, n)/(n*2^n + 1)

(PARI) a(n)=polcoeff(((1-sqrt(1 - 2^(n+3)*x))/2^(n+2)/x)^(1/2^n), n)

Cf. A159558, A159478, A158093, A000108.

Sequence in context: A015215 A158120 A156213 this_sequence A134795 A094680 A061192

Adjacent sequences: A159315 A159316 A159317 this_sequence A159319 A159320 A159321

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 22 2009