A296663 Row sums of A296664.

1, 1, 4, 7, 20, 38, 96, 187, 444, 874, 2000, 3958, 8840, 17548, 38528, 76627, 166124, 330818, 710256, 1415650, 3016056, 6015316, 12736064, 25413342, 53530840, 106853668, 224107936, 447472972, 935062544, 1867450648, 3890018816, 7770342787, 16141765964

Offset: 0

Author

Peter Luschny, Dec 19 2017

Keywords

• nonn

Crossrefs

Cf. A296664, A000531 (bisection).

Programs

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}]

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