A212303 a(n) = n!/([(n-1)/2]!*[(n+1)/2]!) for n>0, a(0)=0, and where [ ] = floor.

0, 1, 2, 3, 12, 10, 60, 35, 280, 126, 1260, 462, 5544, 1716, 24024, 6435, 102960, 24310, 437580, 92378, 1847560, 352716, 7759752, 1352078, 32449872, 5200300, 135207800, 20058300, 561632400, 77558760, 2326762800, 300540195, 9617286240, 1166803110, 39671305740

Offset: 0

Author

Peter Luschny, Oct 24 2013

Keywords

• nonn

Comments

a(n) + A056040(n) = A189911(n), the row sums of the extended Catalan triangle A189231.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Peter Luschny, The lost Catalan numbers.

Crossrefs

Cf. A005430, A001700, A056040, A189911.

Programs

Maple

A212303 := proc(n) if n mod 2 = 0 then n*binomial(n, iquo(n,2))/2 else binomial(n+1, iquo(n,2)+1)/2 fi end: seq(A212303(i), i=0..36);

Mathematica

a[n_?EvenQ] := n*Binomial[n, n/2]/2; a[n_?OddQ] := Binomial[n+1, Quotient[n, 2]+1]/2; Table[a[n], {n, 0, 36}]  (* Jean-François Alcover, Feb 05 2014 *)
nxt[{n_,a_}]:={n+1,If[OddQ[n],a(n+1),(4a(n+1))/(n(n+2))]}; Join[{0}, Transpose[ NestList[ nxt,{1,1},40]][[2]]] (* Harvey P. Dale, Dec 20 2014 *)

Sage

def A212303():
    yield 0
    r, n = 1, 1
    while True:
        yield r
        n += 1
        r *= n if is_even(n) else 4*n/((n-1)*(n+1))
a = A212303(); [next(a) for i in range(36)]

Formula

E.g.f.: (1+x)*BesselI(1, 2*x).

O.g.f.: -((4*x^2-1)^(3/2)+I-(4*I)*x^2+(4*I)*x^3)/(2*x*(4*x^2-1)^(3/2)).

Recurrence: a(n) = n if n < 2 else a(n) = a(n-1)*n if n is even else a(n-1)*n*4/((n-1)*(n+1)).

a(2*n) = n*C(2*n, n) (A005430); a(2*n+1) = C(2*n+1, n+1) (A001700).

a(n) = n$*floor((n+1)/2)^((-1)^n), where n$ is the swinging factorial A056040.

a(n) = Sum_{k=0..n} A189231(n, 2*k+1).

Sum_{n>=1} 1/a(n) = 2/3 + (7/27)*sqrt(3)*Pi.

Sum_{n>=1} (-1)^(n+1)/a(n) = 2/3 + Pi/(9*sqrt(3)). - Amiram Eldar, Aug 20 2022