160619(3)

Ruby

Glaisher's chi_4(n)(3)

ちなみに、さらに整理した
Π(1 - (-x)^n)(1 + x^(2n)) = Π(1 + x^n)(1 - x^(2n))

も確認したが、かえって遅くなってしまった。

# m次以下を取り出す
def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  (0..s1 - 1).each{|i|
    (0..s2 - 1).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary[0..m]
end

def p0(n)
  (3 * n * n - n) / 2
end

def p1(n)
  (3 * n * n + n) / 2
end

def A010815(n)
  ary = Array.new(n + 1, 0)
  ary[0] = 1
  i = 1
  j = p0(i)
  while j <= n
    ary[j] = (-1) ** i
    i += 1
    j = p0(i)
  end
  i = 1
  j = p1(i)
  while j <= n
    ary[j] = (-1) ** i
    i += 1
    j = p1(i)
  end
  ary
end

def A(n)
  ary = A010815(n)
  ary1 = (0..n).map{|i| (-1) ** i * ary[i]}
  ary2 = [1]
  2.step(n, 2){|i|
    b_ary = Array.new(i + 1, 0)
    b_ary[0], b_ary[-1] = 1, 1
    ary2 = mul(ary2, b_ary, n)
  }
  mul(ary1, ary2, n)
end

def B(n)
  ary = A010815(n)
  ary2 = Array.new(n + 1, 0)
  (0..n / 2).each{|i| ary2[2 * i] = ary[i]}
  ary1 = [1]
  (1..n).each{|i|
    b_ary = Array.new(i + 1, 0)
    b_ary[0], b_ary[-1] = 1, 1
    ary1 = mul(ary1, b_ary, n)
  }
  mul(ary1, ary2, n)
end

n = 1000
p a = A(n)
p b = B(n)
p a == b

出力結果
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true