Documentation

Mathlib.Data.Nat.Factorial.SuperFactorial

Superfactorial #

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

Main declarations #

Nat.superFactorial n is the superfactorial of n.

Equations
Instances For

    sf notation for superfactorial

    Equations
    Instances For
      theorem Nat.superFactorial_succ (n : ) :
      n.succ.superFactorial = (n + 1).factorial * n.superFactorial
      @[simp]
      theorem Nat.prod_Icc_factorial (n : ) :
      xFinset.Icc 1 n, x.factorial = n.superFactorial
      @[simp]
      theorem Nat.prod_range_factorial_succ (n : ) :
      xFinset.range n, (x + 1).factorial = n.superFactorial
      @[simp]
      theorem Nat.prod_range_succ_factorial (n : ) :
      xFinset.range (n + 1), x.factorial = n.superFactorial
      theorem Nat.det_vandermonde_id_eq_superFactorial {R : Type u_1} [CommRing R] (n : ) :
      (Matrix.vandermonde fun (i : Fin (n + 1)) => i).det = n.superFactorial
      theorem Nat.superFactorial_two_mul (n : ) :
      (2 * n).superFactorial = (∏ iFinset.range n, (2 * i + 1).factorial) ^ 2 * 2 ^ n * n.factorial
      theorem Nat.superFactorial_four_mul (n : ) :
      (4 * n).superFactorial = ((∏ iFinset.range (2 * n), (2 * i + 1).factorial) * 2 ^ n) ^ 2 * (2 * n).factorial
      theorem Nat.superFactorial_dvd_vandermonde_det {n : } (v : Fin (n + 1)) :
      n.superFactorial (Matrix.vandermonde v).det