Miscellany

(define small-odd-swing
  (vector 1 1 1 3 3 15 5 35 35 315 63 693 231 3003 429 6435 6435 109395
          12155 230945 46189 969969 88179 2028117 676039 16900975 1300075
          35102025 5014575 145422675 9694845 300540195 300540195))

(define (swing n)
  (if (< n 33) (vector-ref small-odd-swing n)
      (let* ((primes (sieve (range 3 n #:by 2)))
             (rootn (integer-sqrt n))
             (primesa (take-while (curryr <= rootn) primes))
             (primesb (filter (λ (x) (odd? (quotient n x)))
                                 (take-while (curryr <= (quotient n 3))
                                             (drop-while (curryr <= rootn) primes)))))
        (apply
         *
         (append
          (map
           (λ (prime)
             (do ((q (quotient n prime) (quotient q prime))
                  (p 1 (if (odd? q) (* p prime) p)))
               ((zero? q) p)))
           primesa)
          (take-while (curryr <= n)
                      (drop-while (curryr <= (quotient n 2)) primes))
          primesb)))))

(define (rec-fact n)
  (if (< n 2) 1
      (* (expt (rec-fact (quotient n 2)) 2) (swing n))))

(define (fact n)
  (if (< n 20) (Π values 1 n)
      (arithmetic-shift (rec-fact n) (- n (bitwise-bit-count n)))))