OFFSET
0,3
FORMULA
a(n) = 2^n*(2*h(n)/sqrt(Pi) - 1) where h(n) = (n/2+1)*Gamma((n+1)/2)/Gamma((n+2)/2) if n mod 2 = 0 else Gamma((n+4)/2)/Gamma((n+3)/2).
a(n) = 2^n*((n+2+(n mod 2))*binomial((n-1+3*(n mod 2))/2, -1/2) - 1).
-(n+1)*(n^2-2*n-1) *a(n) +2*(n-2)*(n^2+n+1) *a(n-1) +4*(n-1)*(n^2-n-5) *a(n-2) -8*(n-2)*(n^2-2) *a(n-3)=0. - R. J. Mathar, Jan 03 2018
MAPLE
a := proc(n) if n mod 2 = 0 then ((n+2)/2)*GAMMA((n+1)/2)/GAMMA((n+2)/2)
else GAMMA((n+4)/2)/GAMMA((n+3)/2) fi; 2^n*(2*%/sqrt(Pi)-1) end:
seq(a(n), n=0..32);
MATHEMATICA
a[n_] := 2^n ((n + 2 + Mod[n, 2]) Binomial[(n - 1 + 3 Mod[n, 2])/2, -1/2] - 1);
Table[a[n], {n, 0, 32}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Dec 19 2017
STATUS
approved