OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Luschny, The lost Catalan numbers.
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(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
MAPLE
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 *)
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Oct 24 2013
STATUS
approved