A266971 Expansion of Product_{k>=1} 1 / (1 + k*x^k)^k.

1, -1, -3, -6, 2, 9, 41, 46, 91, -110, -210, -713, -574, -1152, 792, 1066, 9317, 8553, 21302, 745, 8051, -82940, -76750, -276022, -82369, -404100, 381095, -38110, 2427272, 1126260, 6527840, 198507, 9754305, -14320206, 2879362, -60271740, -5154261, -143468194

Offset: 0

Author

Vaclav Kotesovec, Jan 07 2016

Keywords

• sign

Comments

For n > 36 is a(n) > 0 if n is even and a(n) < 0 if n is odd.

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n, g(n) = -n. - Seiichi Manyama, Nov 18 2017

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..6224 (terms 0..1000 from Vaclav Kotesovec)

Crossrefs

Cf. A022629, A022693, A266891, A266941, A266964.

Programs

Mathematica

nmax=50; CoefficientList[Series[Product[1/(1+k*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

PARI

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+k*x^k)^k)) \\ Seiichi Manyama, Nov 18 2017

Ruby

def s(f_ary, g_ary, n)
  s = 0
  (1..n).each{|i| s += i * f_ary[i] * g_ary[i] ** (n / i) if n % i == 0}
  s
end
def A(f_ary, g_ary, n)
  ary = [1]
  a = [0] + (1..n).map{|i| s(f_ary, g_ary, i)}
  (1..n).each{|i| ary << (1..i).inject(0){|s, j| s + a[j] * ary[-j]} / i}
  ary
end
def A266971(n)
  A((0..n).to_a, (0..n).map{|i| -i}, n)
end
p A266971(50) # Seiichi Manyama, Nov 18 2017

Formula

a(n) ~ c * (-1)^n * n^2 * 3^(n/3), where

c = 50.5838262902886367070621... if mod(n,3)=0,

c = 50.5827771239052189170531... if mod(n,3)=1,

c = 50.5832885870455104598393... if mod(n,3)=2.

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(-d)^(n/d). - Seiichi Manyama, Nov 18 2017