cp's OEIS Frontend

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

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