have : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (sqrt π)) := by
  refine' Tendsto.comp Stirling.tendsto_stirlingSeq_sqrt_pi _
  exact tendsto_id.const_mul_atTop' two_pos
this✝ : 0 < π
this : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
⊢ Tendsto (fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) atTop (𝓝 (Real.sqrt π / Real.sqrt π ^ 2))
this✝¹ : 0 < π
this✝ : (Real.sqrt π)⁻¹ = Real.sqrt π / Real.sqrt π ^ 2
this : Tendsto (fun n => Stirling.stirlingSeq (2 * n)) atTop (𝓝 (Real.sqrt π))
⊢ Tendsto (fun n => ↑(centralBinom n) * Real.sqrt ↑n / 4 ^ n) atTop (𝓝 (Real.sqrt π / Real.sqrt π ^ 2))