A238452 Second column of the extended Catalan triangle A189231.

0, 1, 2, 2, 8, 5, 30, 14, 112, 42, 420, 132, 1584, 429, 6006, 1430, 22880, 4862, 87516, 16796, 335920, 58786, 1293292, 208012, 4992288, 742900, 19315400, 2674440, 74884320, 9694845, 290845350, 35357670, 1131445440, 129644790, 4407922860, 477638700, 17194993200

Offset: 0

Author

Peter Luschny, Mar 01 2014

Keywords

• nonn
• easy

Crossrefs

Cf. A000108, A057977, A162551, A189231.

Programs

Maple

a := proc(n) option remember;
  if n < 3 then return n fi;
  if n mod 2 = 0 then return n*a(n-1) fi;
  h := iquo(n,2); n*a(n-1)/(h*(h+2)) end:
seq(a(n), n=0..36);

Mathematica

t[n_, k_] /; (k > n || k < 0) = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] =
  t[n - 1, k - 1] + Mod[n - k, 2] t[n - 1, k] + t[n - 1, k + 1];
a[n_] := t[n, 1];
Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Jul 10 2019 *)

Sage

def A238452():
    a = 1; n = 2
    yield 0
    while True:
        yield a
        a *= n
        if is_odd(n):
            a /= (n//2*(n//2+2))
        n += 1
a = A238452(); [next(a) for n in range(36)]

Formula

Definition: a(n) = binomial(n+1, floor(n/2)+1) / (floor(n/2)+2) if n is odd, and 2*binomial(n, floor(n/2)+1) otherwise.

a(n) = A189231(n, 1).

a(n) = A238762(n+1, n-1).

a(2*n) = A162551(n).

a(2*n+1) = A000108(n+1).

a(n) = A057977(n+1) - A057977(n)*((n+1) mod 2). - Peter Luschny, Aug 07 2016