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Bug in infinite sum

Consider this nice sum

s[x_] = Sum[Binomial[x, k], {k, 0, Infinity}]

and that, (with the odd terms only)

u[x_] = Sum[Binomial[x, 2*k + 1], {k, 0, Infinity}]
2^(-1 + x)

But now the even terms only ... and the surprise:
g[x_] = Sum[Binomial[x, 2*k], {k, 0, Infinity}]

(Sqrt[Pi]*Gamma[1 + x]*GegenbauerC[x, 1/2 - x, 1])/
  (2^x*Gamma[1/2 + x])

looks complicated, but let's see
FullSimplify[g[x], x > 0]


much simpler, but definitely wrong (giving e.g. 0 for x=1/2)
Of course g should be s - u = 2^x -1/2 2^s = 1/2 2^s.

Best regards,

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