$Assumptions=M>m>0&& M∈Integers&&m∈integers;

K[s_]:=Log[(((((1+Exp[s])/2)))^M)]
K1[s_]=D[K[s],s]
K2[s_]=D[K[s],{s,2}]
K3[s_]=D[K[s],{s,3}]
K4[s_]=D[K[s],{s,4}]
K5[s_]=D[K[s],{s,5}]
K6[s_]=D[K[s],{s,6}]
K1[0]
K2[0]
K3[0]
K4[0]
K5[0]
K6[0]

K[s_]:=Log[(((((1+Exp[s/M])/2)))^M)]
K1[s_]=D[K[s],s]
K2[s_]=D[K[s],{s,2}]
K3[s_]=D[K[s],{s,3}]
K4[s_]=D[K[s],{s,4}]
K5[s_]=D[K[s],{s,5}]
K6[s_]=D[K[s],{s,6}]
K1[0]
K2[0]
K3[0]
K4[0]
K5[0]
K6[0]

Clear[M]
W[m_]:=Binomial[M,m]
D[Log[W[m]],m]
Sum[W[m],{m, 0, M}]

Clear[M]
M=150
lnW[x_]:=M Log[(M/(M-x))]-m Log[(x/(M-x))]
Wstir[x_]:=(((((M/(M-x))))^M)/((((x/(M-x))))^x))(1/(2^M))
(*Integrate[Wstir[x],{x,0,M}]*)
NIntegrate[Wstir[x],{x,0,M}]
N[((Sqrt[(2/(π M))])^(-1))]