Documentation

Mathlib.Data.Nat.Factorial.SuperFactorial

Superfactorial #

This file defines the superfactorial sf n = 1! * 2! * 3! * ... * n!.

Main declarations #

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def Nat.superFactorial :
ℕ → ℕ

Nat.superFactorial n is the superfactorial of n.

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    Instances For
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      def Nat.termSf_ :
      Lean.ParserDescr

      sf notation for superfactorial

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        Instances For
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          @[simp]
          theorem Nat.superFactorial_zero :
          superFactorial 0 = 1
          source
          theorem Nat.superFactorial_succ (n : ℕ) :
          n.succ.superFactorial = (n + 1).factorial * n.superFactorial
          source
          @[simp]
          theorem Nat.superFactorial_one :
          superFactorial 1 = 1
          source
          @[simp]
          theorem Nat.superFactorial_two :
          superFactorial 2 = 2
          source
          @[simp]
          theorem Nat.prod_Icc_factorial (n : ℕ) :
          ∏ x ∈ Finset.Icc 1 n, x.factorial = n.superFactorial
          source
          @[simp]
          theorem Nat.prod_range_factorial_succ (n : ℕ) :
          ∏ x ∈ Finset.range n, (x + 1).factorial = n.superFactorial
          source
          @[simp]
          theorem Nat.prod_range_succ_factorial (n : ℕ) :
          ∏ x ∈ Finset.range (n + 1), x.factorial = n.superFactorial
          source
          theorem Nat.det_vandermonde_id_eq_superFactorial {R : Type u_1} [CommRing R] (n : ℕ) :
          (Matrix.vandermonde fun (i : Fin (n + 1)) => ↑↑i).det = ↑n.superFactorial
          source
          theorem Nat.superFactorial_two_mul (n : ℕ) :
          (2 * n).superFactorial = (∏ i ∈ Finset.range n, (2 * i + 1).factorial) ^ 2 * 2 ^ n * n.factorial
          source
          theorem Nat.superFactorial_four_mul (n : ℕ) :
          (4 * n).superFactorial = ((∏ i ∈ Finset.range (2 * n), (2 * i + 1).factorial) * 2 ^ n) ^ 2 * (2 * n).factorial
          source
          theorem Nat.superFactorial_dvd_vandermonde_det {n : ℕ} (v : Fin (n + 1) → ℤ) :
          ↑n.superFactorial ∣ (Matrix.vandermonde v).det