Relation between binomial (2n,n), binomial(1/2,n) and binomial (-1/2,n)

> f:= proc(n) binomial(2*n,n); end proc;

f := proc(n) ... end;

> g := proc(n) binomial(-1/2,n)*(-4)^n; end proc;

g := proc(n) ... end;

> h := proc(n) binomial(1/2,n)*(2*n-1)*(4)^n *(-1)^(n-1); end proc;

h := proc(n) ... end;

> seq([f(n),g(n),h(n)],n =0..20);

[1, 1, 1], [2, 2, 2], [6, 6, 6], [20, 20, 20], [70, 70, 70], [252, 252, 252], [924, 924, 924], [3432, 3432, 3432], [12870, 12870, 12870], [48620, 48620, 48620], [184756, 184756, 184756], [705432, 705432, 705432], [2704156, 2704156, 2704156], [10400600, 10400600, 10400600], [40116600, 40116600, 40116600], [155117520, 155117520, 155117520], [601080390, 601080390, 601080390], [2333606220, 2333606220, 2333606220], [9075135300, 9075135300, 9075135300],[35345263800, 35345263800, 35345263800], [137846528820, 137846528820, 137846528820]

Thus its easy to see that we can go back and forth between biomial (2n,n), binomial(1/2,n) and binomial(-1/2,n). Note the extra (-1)*(2n-1) term for binomial(1/2,n) over binomial(-1/2,n)