DoubleFactorialNumbers

The odd double factorials $1, 3, 15, 105, …$ — $(2n-1)!!$ — perfect matchings of $K_{2n}$.

OEISA001147↗Wolfram LanguageFactorial2↗mathlib4Nat.doubleFactorial↗

DoubleFactorialNumbers$(2n-1)!!$ for $n = 1, 2, 3, …$.

Domain: Collections

Details

• A lazy indexed collection: $Count(DoubleFactorialNumbers) = +\infty$, and $At(DoubleFactorialNumbers, k) = (2k-1)!!$, computed exactly over arbitrary-precision integers -- $At(DoubleFactorialNumbers, 5) = 945$.
• OEIS A001147, whose offset differs from ours the same way as FactorialNumbers: A001147 opens with the duplicate $a(0) = a(1) = 1$; this collection starts at $n=1$ without repeating it.
• Membership goes through Element by growing the sequence forward until it reaches or passes the candidate -- $Element(945, DoubleFactorialNumbers)$ is true, $Element(900, DoubleFactorialNumbers)$ is false.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

Fold(Function([wk_w[1] * (2wk_n + 3)], wk_w, wk_n), [1], Range(0, _r - 1, 1))[1]

rank(_x)

Fold(
      Function(
                wk_w if wk_n >= 0 && wk_w[1] > _x else [
                                                         wk_w[1] * (2wk_n + 3),
                                                         wk_n if wk_w[2] < 0 &&
                                                                 wk_w[1] == _x else wk_w[2],
                                                       ],
                wk_w,
                wk_n,
              ),
      [1, -1],
      Range(
             0,
             1 * (Apply(
                         (fl_r: list<integer>) => fl_r[2],
                         Fold(Function(
                                        [
                                          (fl_s[1] - fl_s[1] % 2 ^ fl_k) / 2 ^
                                                                           fl_k,
                                          fl_s[2] + fl_k,
                                        ] if fl_s[1] >= 2 ^ fl_k else fl_s,
                                        fl_s,
                                        fl_k,
                                      ), [_x + 2, 0], [
                                                       8_192,
                                                       4_096,
                                                       2_048,
                                                       1_024,
                                                       512,
                                                       256,
                                                       128,
                                                       64,
                                                       32,
                                                       16,
                                                       8,
                                                       4,
                                                       2,
                                                       1,
                                                     ]),
                       ) + 1) + 12,
             1,
           ),
    )[2] if _x >= 0 else -1

valid(_x)

Fold(
      Function(
                wk_w if wk_n >= 0 && wk_w[1] > _x else [
                                                         wk_w[1] * (2wk_n + 3),
                                                         wk_n if wk_w[2] < 0 &&
                                                                 wk_w[1] == _x else wk_w[2],
                                                       ],
                wk_w,
                wk_n,
              ),
      [1, -1],
      Range(
             0,
             1 * (Apply(
                         (fl_r: list<integer>) => fl_r[2],
                         Fold(Function(
                                        [
                                          (fl_s[1] - fl_s[1] % 2 ^ fl_k) / 2 ^
                                                                           fl_k,
                                          fl_s[2] + fl_k,
                                        ] if fl_s[1] >= 2 ^ fl_k else fl_s,
                                        fl_s,
                                        fl_k,
                                      ), [_x + 2, 0], [
                                                       8_192,
                                                       4_096,
                                                       2_048,
                                                       1_024,
                                                       512,
                                                       256,
                                                       128,
                                                       64,
                                                       32,
                                                       16,
                                                       8,
                                                       4,
                                                       2,
                                                       1,
                                                     ]),
                       ) + 1) + 12,
             1,
           ),
    )[2] >= 0 if _x >= 0 else False

Enumeration

Examples

1 more kept as data, checked against the oracles

About

Examples	1
Crosswalk	OEISWolfram Languagemathlib4

See also: FactorialNumbers, PhylogeneticTrees, Count, At, Element