Central binomial coefficients and harmonic sums

Finite sums and power series involving central binomial coefficients or their reciprocals, often weighted by harmonic or odd harmonic numbers, including arcsin-type series and Kollár's incomplete binomial sums.

DefinitionAerated central binomial numbers#✓fully proved

For a natural number \(n\), the \(n\)-th aerated central binomial number is \(\binom{n}{n/2}\) if \(n\) is even and \(0\) if \(n\) is odd. The sequence begins \(1, 0, 2, 0, 6, 0, 20, \dots\).

def aeratedCentralBinomialNumber (n : ℕ) : ℕ :=
  if n % 2 = 0 then Nat.choose n (n / 2) else 0

TheoremCentral binomial power sums via Stirling numbers#✓fully proved

For every natural number \(m\) and every real \(x\) with \(|x| < 1/4\),

\[\sum_{n=0}^{\infty} \binom{2n}{n} n^m x^n = \frac{1}{\sqrt{1-4x}} \sum_{k=0}^{m} S(m,k) \binom{2k}{k}\, k! \left(\frac{x}{1-4x}\right)^k,\]

where \(S(m,k)\) denotes the Stirling numbers of the second kind (and \(n^0 = 1\) for every \(n\)).

The source asks for positive \(m\); the identity also holds at \(m = 0\), where both sides are the generating function of the central binomial coefficients.

theorem central_binom_power_sum_stirling
    (m : ℕ) (x : ℝ) (hx : |x| < 1 / 4) :
    ∑' n : ℕ, (Nat.centralBinom n : ℝ) * (n : ℝ) ^ m * x ^ n =
      1 / Real.sqrt (1 - 4 * x) *
        ∑ k ∈ Finset.range (m + 1),
          (Nat.stirlingSecond m k : ℝ) * (Nat.centralBinom k : ℝ) *
            (Nat.factorial k : ℝ) * (x / (1 - 4 * x)) ^ k
Sources
  • Khristo N. Boyadzhiev, "Series with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers," Journal of Integer Sequences 15 (2012), Article 12.1.7, Theorem (equation 8), lines 155–160, https://cs.uwaterloo.ca/journals/JIS/VOL15/Boyadzhiev/boyadzhiev6.tex
2 fully proved

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.