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What's about substituting n to make it an odd integer?

var("s k n t")
assume(k, 'integer')    
assume(k>0)    
term= sqrt(s/n)*1/sqrt(2*pi)*exp(-s*t^2/(2*n))*exp(-s/2)*s^(n/2-1)/(gamma(n/2)*2^(n/2))    
tdist=integral(term.substitute(n=2*k+1),s,0,infinity)    
tdist