A303307 Expansion of Product_{n>=1} ((1 + (2*x)^n)/(1 - (2*x)^n))^(1/2).

1, 2, 6, 20, 54, 156, 444, 1192, 3174, 8620, 22516, 58392, 151996, 387352, 984888, 2507088, 6270854, 15659724, 39067588, 96454072, 237663444, 584266696, 1425921992, 3470869296, 8431325916, 20380759544, 49122457608, 118178637040, 283150466232, 676768288176

Offset: 0

Author

Seiichi Manyama, Apr 21 2018

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3000

Crossrefs

Cf. A054785, A303306, A303342.

Programs

Mathematica

CoefficientList[Series[Sqrt[QPochhammer[-1, 2*x] / (2*QPochhammer[2*x])], {x, 0, 30}], x] (* Vaclav Kotesovec, Apr 21 2018 *)

Ruby

def s(n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0}
  s
end
def A303307(n)
  ary = [1]
  a = (0..n).map{|i| 2 ** (i - 1) * (s(2 * i) - s(i))}
  (1..n).each{|i| ary << (1..i).inject(0){|s, j| s + a[j] * ary[-j]} / i}
  ary
end
p A303307(100)

Formula

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} 2^(k-1) * A054785(k) * a(n-k) for n > 0.

a(n) ~ 2^(n - 21/8) * exp(Pi*sqrt(n/2)) / n^(7/8). - Vaclav Kotesovec, Apr 21 2018