A296663 - OEIS

%I #10 Jan 03 2018 07:39:08

%S 1,1,4,7,20,38,96,187,444,874,2000,3958,8840,17548,38528,76627,166124,

%T 330818,710256,1415650,3016056,6015316,12736064,25413342,53530840,

%U 106853668,224107936,447472972,935062544,1867450648,3890018816,7770342787,16141765964

%N Row sums of A296664.

%F 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).

%F a(n) = 2^n*((n+2+(n mod 2))*binomial((n-1+3*(n mod 2))/2, -1/2) - 1).

%F -(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

%p a := proc(n) if n mod 2 = 0 then ((n+2)/2)*GAMMA((n+1)/2)/GAMMA((n+2)/2)

%p else GAMMA((n+4)/2)/GAMMA((n+3)/2) fi; 2^n*(2*%/sqrt(Pi)-1) end:

%p seq(a(n), n=0..32);

%t a[n_] := 2^n ((n + 2 + Mod[n, 2]) Binomial[(n - 1 + 3 Mod[n, 2])/2, -1/2] - 1);

%t Table[a[n], {n, 0, 32}]

%Y Cf. A296664, A000531 (bisection).

%K nonn

%O 0,3

%A _Peter Luschny_, Dec 19 2017