%I #28 Aug 20 2022 08:05:04
%S 0,1,2,3,12,10,60,35,280,126,1260,462,5544,1716,24024,6435,102960,
%T 24310,437580,92378,1847560,352716,7759752,1352078,32449872,5200300,
%U 135207800,20058300,561632400,77558760,2326762800,300540195,9617286240,1166803110,39671305740
%N a(n) = n!/([(n-1)/2]!*[(n+1)/2]!) for n>0, a(0)=0, and where [ ] = floor.
%C a(n) + A056040(n) = A189911(n), the row sums of the extended Catalan triangle A189231.
%H Vincenzo Librandi, <a href="/A212303/b212303.txt">Table of n, a(n) for n = 0..1000</a>
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>.
%F E.g.f.: (1+x)*BesselI(1, 2*x).
%F 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)).
%F 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)).
%F a(2*n) = n*C(2*n, n) (A005430); a(2*n+1) = C(2*n+1, n+1) (A001700).
%F a(n) = n$*floor((n+1)/2)^((-1)^n), where n$ is the swinging factorial A056040.
%F a(n) = Sum_{k=0..n} A189231(n, 2*k+1).
%F Sum_{n>=1} 1/a(n) = 2/3 + (7/27)*sqrt(3)*Pi.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2/3 + Pi/(9*sqrt(3)). - _Amiram Eldar_, Aug 20 2022
%p 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);
%t 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 *)
%t 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 *)
%o (Sage)
%o def A212303():
%o yield 0
%o r, n = 1, 1
%o while True:
%o yield r
%o n += 1
%o r *= n if is_even(n) else 4*n/((n-1)*(n+1))
%o a = A212303(); [next(a) for i in range(36)]
%Y Cf. A005430, A001700, A056040, A189911.
%K nonn
%O 0,3
%A _Peter Luschny_, Oct 24 2013