それほどでもなかった.
g(x,y) := (-1)*Σ[k=1..n] C{n,k}(-xy)^k /kk
∂[x]∂[y]g(x,y) = Σ[k=1..n] C{n,k}(-xy)^{k-1}
　= (1- (1-xy)^n) / xy　= (1- (1-xy)^n) / (1- (1-xy))
　= Σ[k=0..n-1] (1-xy)^k
∂[x]g(x,y) = ∫dy ... = Σ[k=1..n] (1-(1-xy)^k) / kx　{∵ ∂[x]g(x,0)=0}
　= Σ[k=1..n] y(1-(1-xy)^k) / k(1-(1-xy))
　= Σ[k=1..n] Σ[m=0..k-1] y(1-xy)^m / k
g(x,y) = ∫dx ... = Σ[k=1..n] Σ[m=1..k] { 1- (1-xy)^m }/mk　{∵ g(0,y)=0}

∴ Σ[k=1..n] C{n,k}(-1)^{k+1} /kk
　= g(1,1) = Σ[k=1..n] Σ[m=1..k] 1/mk