OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..58
FORMULA
G.f.: Sum_{n>=0} (-2)^n * log(1 - x/2^n)^n/n! = Sum_{n>=0} a(n)*x^n/2^(n^2).
a(n) = (2^n/n!) * Product_{k=0..n-1} (2^(n-1)*k + 1).
a(n) = [x^n] 1/(1 - 2^n*x)^(2/2^n).
EXAMPLE
G.f.: A(x) = 1 + 2*x/2 + 6*x^2/2^4 + 60*x^3/2^9 + 2550*x^4/2^16 + 476476*x^5/2^25 +...+ a(n)*x^n/2^(n^2) +...
where
A(x) = 1 - 2*log(1-x/2) + 4*log(1-x/4)^2/2! - 8*log(1-x/8)^3/3! + 16*log(1-x/16)^4/4! +...+ (-2)^n*log(1-x/2^n)^n/n! +...
Illustrate a(n) = [x^n] 1/(1 - 2^n*x)^(2/2^n):
(1-x)^(-2/1) = (1) + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 +...
(1-2*x)^(-2/2) = 1 + (2)*x + 4*x^2 + 8*x^3 + 16*x^4 + 32*x^5 +...
(1-4*x)^(-2/4) = 1 + 2*x + (6)*x^2 + 20*x^3 + 70*x^4 + 252*x^5 +...
(1-8*x)^(-2/8) = 1 + 2*x + 10*x^2 + (60)*x^3 + 390*x^4 + 2652*x^5 +...
(1-16*x)^(-2/16) = 1 + 2*x + 18*x^2 + 204*x^3 + (2550)*x^4 + 33660*x^5 +...
(1-32*x)^(-2/32) = 1 + 2*x + 34*x^2 + 748*x^3 + 18326*x^4 + (476476)*x^5 +...
where the coefficients in parenthesis form the initial terms of this sequence.
Particular values.
A(1) = 1 + 2*log(2) + 4*log(4/3)^2/2! + 8*log(8/7)^3/3! + 16*log(16/15)^4/4! +...
A(1/2) = 1 + 2*log(4/3) + 4*log(8/7)^2/2! + 8*log(16/15)^3/3! +...
A(1/4) = 1 + 2*log(8/7) + 4*log(16/15)^2/2! + 8*log(32/31)^3/3! +...
A(3/2) = 1 + 2*log(4) + 4*log(8/5)^2/2! + 8*log(16/13)^3/3! + 16*log(32/29)^4/4! +...
Explicitly,
A(1) = 2.55500248436101360804704969796239525102504151...
A(1/2) = 1.61138451105646219391156983544059555709337920...
A(1/4) = 1.27543593708175757392940597050033002345086132...
A(3/2) = 4.22639446385430649517540615961613624264078875...
MATHEMATICA
Table[2^(n^2) Binomial[n-1+1/2^(n-1), n], {n, 0, 20}] (* Harvey P. Dale, Feb 01 2017 *)
PROG
(PARI) {a(n)=2^(n^2)*binomial(n-1+1/2^(n-1), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=(2^n/n!)*prod(k=0, n-1, 2^(n-1)*k + 1)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=2^(n^2)*polcoef(sum(k=0, n, (-2)^k*log(1-x/2^k +x*O(x^n))^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 23 2013
STATUS
approved