SchroederNumbers

The (large) Schröder numbers $1, 2, 6, 22, 90, 394, …$ as a lazy indexed collection.

OEISA006318

SchroederNumbers

S_0 = 1

S_n = 2 s_n

for

n \ge 1

(twice the little Schröder numbers), an infinite indexed collection.

Domain: Collections

Details

A lazy indexed collection: $Count(SchroederNumbers) = +\infty$ , and $At(SchroederNumbers, k)$ unranks term $k-1$ by doubling LittleSchroderNumbers rather than re-deriving a recurrence -- $At(SchroederNumbers, 1) = S_0 = 1$ .
OEIS A006318, starting exactly at its offset-0 term: $1, 2, 6, 22, 90, 394, 1806, …$ ; counts monotone lattice paths from $(0,0)$ to $(n,n)$ using steps $(1,0)$ , $(0,1)$ and $(1,1)$ that never rise above the diagonal.
Membership goes through Element: $Element(22, SchroederNumbers)$ is true, $Element(10, SchroederNumbers)$ is false.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

Fold(
      Function(sm_a + Binomial(_r + sm_k, _r - sm_k) * (Binomial(2sm_k, sm_k) -
                                                        (0 if 2sm_k < sm_k + 1 else Binomial(2sm_k, sm_k + 1))), sm_a, sm_k),
      0,
      Range(0, _r, 1),
    )

rank(_x)

Apply(
       (sq_at: integer) => sq_at if Fold(
                                          Function(sm_a + Binomial(
                                                                    sq_at + sm_k,
                                                                    sq_at - sm_k,
                                                                  ) * (Binomial(2sm_k, sm_k) -
                                                                      (0 if 2sm_k <
                                                                            sm_k +
                                                                            1 else Binomial(2sm_k, sm_k + 1))), sm_a, sm_k),
                                          0,
                                          Range(0, sq_at, 1),
                                        ) == _x else -1,
       Fold(
             Function(
                       sq_at + 2 ^ sq_b if Fold(
                                                 Function(sm_a + Binomial(sq_at +
                                                                          2 ^
                                                                          sq_b +
                                                                          sm_k, sq_at +
                                                                            2 ^
                                                                            sq_b -
                                                                            sm_k) *
                                                                 (Binomial(
                                                                            2sm_k,
                                                                            sm_k,
                                                                          ) -
                                                                  (0 if 2sm_k <
                                                                        sm_k + 1 else Binomial(2sm_k, sm_k + 1))), sm_a, sm_k),
                                                 0,
                                                 Range(0, sq_at + 2 ^ sq_b, 1),
                                               ) <= _x else sq_at,
                       sq_at,
                       sq_b,
                     ),
             0,
             Range(Floor(Log(Floor(Log(_x, 2)) + 2, 2)) + 1, 0, -1),
           ),
     ) if True else -1

valid(_x)

Apply(
       (sq_at: integer) => Fold(
                                 Function(
                                           sm_a + Binomial(
                                                            sq_at + sm_k,
                                                            sq_at - sm_k,
                                                          ) * (Binomial(
                                                                        2sm_k,
                                                                        sm_k,
                                                                      ) -
                                                              (0 if 2sm_k < sm_k +
                                                                            1 else Binomial(2sm_k, sm_k + 1))),
                                           sm_a,
                                           sm_k,
                                         ),
                                 0,
                                 Range(0, sq_at, 1),
                               ) == _x,
       Fold(
             Function(
                       sq_at + 2 ^ sq_b if Fold(
                                                 Function(sm_a + Binomial(sq_at +
                                                                          2 ^
                                                                          sq_b +
                                                                          sm_k, sq_at +
                                                                            2 ^
                                                                            sq_b -
                                                                            sm_k) *
                                                                 (Binomial(
                                                                            2sm_k,
                                                                            sm_k,
                                                                          ) -
                                                                  (0 if 2sm_k <
                                                                        sm_k + 1 else Binomial(2sm_k, sm_k + 1))), sm_a, sm_k),
                                                 0,
                                                 Range(0, sq_at + 2 ^ sq_b, 1),
                                               ) <= _x else sq_at,
                       sq_at,
                       sq_b,
                     ),
             0,
             Range(Floor(Log(Floor(Log(_x, 2)) + 2, 2)) + 1, 0, -1),
           ),
     ) if True && _x >= 1 else False

Enumeration

Examples

1 more kept as data, checked against the oracles

About

Examples	1
Crosswalk	OEIS

SchroederNumbers ​

Definition

Enumeration

Examples

SchroederNumbers