A303343 -id:A303343 - cp's OEIS frontend

A303306 Expansion of Product_{n>=1} ((1 - (2x)^n)/(1 + (2x)^n))^(1/2).

1, -2, -2, -4, 6, 4, 12, 56, 134, -108, 196, 328, -484, -88, -3752, -18576, 16838, -16460, -95340, -24408, -201036, -472584, 565544, 1424144, 1843356, -6632568, 10365224, 2317008, 49620088, 130484688, -4419664, 631241440, 761908550, -29690892, 329427380, -8889717144, 23673793860

Offset: 0

Seiichi Manyama, Apr 21 2018

sign

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

Cf. A054785, A303307, A303343.

def s(n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0}
  s
end
def A303306(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 A303306(100)

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

A303344 Expansion of Product_{n>=1} ((1 + (nx)^n)/(1 - (nx)^n))^(1/n).

Original entry on oeis.org

1, 2, 6, 28, 182, 1640, 19220, 278224, 4809942, 96598622, 2208156512, 56580566908, 1605518324884, 49963000166616, 1691615823420800, 61897541544248720, 2433873670903995990, 102341746590575878628, 4582360425862350559350, 217661837260679635780356

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

Cf. A023881, A186633, A303307, A303343.

nmax = 20; CoefficientList[Series[Product[((1 + (k*x)^k)/(1 - (k*x)^k))^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+(k*x)^k)/(1-(k*x)^k))^(1/k)))

a(n) ~ 2 * n^(n-1). - Vaclav Kotesovec, Apr 22 2018

G.f.: exp(Sum_{k>=1} (sigma_k(2*k) - sigma_k(k))*x^k/(2^(k-1)*k)). - Ilya Gutkovskiy, Apr 14 2019

cp's OEIS Frontend

A303306 Expansion of Product_{n>=1} ((1 - (2x)^n)/(1 + (2x)^n))^(1/2).

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A303344 Expansion of Product_{n>=1} ((1 + (nx)^n)/(1 - (nx)^n))^(1/n).

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cp's OEIS Frontend

A303306 Expansion of Product_{n>=1} ((1 - (2*x)^n)/(1 + (2*x)^n))^(1/2).

Views

Author

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Programs

Ruby

Formula

A303344 Expansion of Product_{n>=1} ((1 + (n*x)^n)/(1 - (n*x)^n))^(1/n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A303306 Expansion of Product_{n>=1} ((1 - (2x)^n)/(1 + (2x)^n))^(1/2).

A303344 Expansion of Product_{n>=1} ((1 + (nx)^n)/(1 - (nx)^n))^(1/n).