0,2

Sum_{n>=0} C(1/2^n, n) = 1.4306345243611686570661803375590...

G.f.: Sum_{n>=0} a(n)*x^n/2^(n^2+n) = Sum_{n>=0} log(1 + x/2^n)^n/n!.

a(n) = [x^n] (1 + 2^(n+1)*x)^(1/2^n).

G.f.: A(x) = 1 +2*x/2^2 -6*x^2/2^6 +140*x^3/2^12 -14570*x^4/2^20 +...

A(x) = 1 + log(1+x/2) + log(1+x/4)^2/2! + log(1+x/8)^3/3! +...

Illustrate a(n) = [x^n] (1 + 2^(n+1)*x)^(1/2^n):

(1+4*x)^(1/2) = 1 + (2)*x - 2*x^2 + 4*x^3 - 10*x^4 +...

(1+8*x)^(1/4) = 1 + 2*x - (6)*x^2 + 28*x^3 - 154*x^4 +...

(1+16*x)^(1/8) = 1 + 2*x - 14*x^2 + (140)*x^3 - 1610*x^4 +...

(1+32*x)^(1/16) = 1 + 2*x - 30*x^2 + 620*x^3 - (14570)*x^4 +...

(1+64*x)^(1/32) = 1 + 2*x - 62*x^2 + 2604*x^3 - 123690*x^4 + (6283452)*x^5 +...

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

Sequence in context: A101753 A156515 A090907 this_sequence A047937 A027731 A145143

Adjacent sequences: A159475 A159476 A159477 this_sequence A159479 A159480 A159481

sign

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 19 2009