Median Genocchi numbers

DefinitionGenocchi numbers via Bernoulli numbers#✓fully proved

For a natural number \(n\), the Genocchi number \(G_n\) is the rational number \(G_n = 2(1 - 2^n) B_n\), where \(B_n\) is the \(n\)-th Bernoulli number with the convention \(B_1 = -\tfrac{1}{2}\).

def genocchiNumberViaBernoulli (n : ℕ) : ℚ :=
  2 * (1 - (2 : ℚ) ^ n) * bernoulli n
Used by 3
Used byOdd-indexed median Genocchi numbers · Divisibility of the median Genocchi numbers · The two definitions of the Genocchi numbers agree

DefinitionOdd-indexed median Genocchi numbers#✓fully proved

For a natural number \(m\), let \(B_m\) be the \(m\)-th Bernoulli number and \(G_m = 2(1 - 2^m) B_m\) the \(m\)-th Genocchi number. For a natural number \(n\), the median Genocchi number \(H_{2n+1}\) is the rational number defined by \(H_1 = 1\) and, for \(n \ge 1\),

\[H_{2n+1} = \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \binom{n}{2k+1} G_{2n-2k}.\]

These values carry signs: \(H_3 = -1\) and \(H_5 = 2\).

def medianGenocchiOdd (n : ℕ) : ℚ :=
  if n = 0 then 1
  else ∑ k ∈ Finset.range ((n - 1) / 2 + 1),
    (Nat.choose n (2 * k + 1) : ℚ) * genocchiNumberViaBernoulli (2 * n - 2 * k)
Uses 1 · Used by 1
Statement usesGenocchi numbers via Bernoulli numbers
Used byDivisibility of the median Genocchi numbers

TheoremDivisibility of the median Genocchi numbers#✓fully proved

Let \(H_{2m+1}\) denote the odd-indexed median Genocchi values, with the signs given by their defining sum, so that \(H_5 = 2\), \(H_7 = -8\) and \(H_9 = 56\). For every natural number \(n \ge 1\) there is an integer \(z\) with \(H_{2n+3} = 2^n z\), and \(z \equiv 1 \pmod 6\) if \(n\) is odd, while \(z \equiv 4 \pmod 6\) if \(n\) is even.

Proof: The antisymmetry of the Euler–Seidel matrix of the Genocchi numbers writes \(H_{2n+3}\) as an alternating sum of Legendre–Stirling numbers times \(k!\,(k+1)!\). Pulling a power of two out of each term gives the factor \(2^n\); in the remaining sum every term from the fourth on is divisible by \(6\), which gives the residue. The divisibility and residue arguments are Chen's, applied to this finite sum instead of his continued fraction.

theorem medianGenocchiOdd_divisibility (n : ℕ) (hn : 1 ≤ n) :
    ∃ z : ℤ, medianGenocchiOdd (n + 1) = (2 : ℚ) ^ n * z ∧
      Int.ModEq 6 z (if Odd n then 1 else 4)
Sources
Uses 9
Statement usesOdd-indexed median Genocchi numbers
Proof usesEuler–Seidel matrix · Closed form of the Euler–Seidel matrix · Genocchi numbers · Genocchi numbers via Bernoulli numbers · The two definitions of the Genocchi numbers agree · Genocchi numbers from their generating function · Generating function of the Genocchi numbers · The Genocchi power series is \(2x/(e^x + 1)\)
3 fully proved

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