On pose A la première somme
A= sum(i=0...n-1) sum(j=i+1...n) j C(n,i)
A= sum(i=0...n-1) C(n,i) sum(j=i+1...n) j
A= sum(i=0...n-1) C(n,i) ( n(n+1)-i(i+1))/2
A= n(n+1)/2. sum(i=0...n-1) C(n,i) - sum(i=0...n-1) C(n,i).i(i+1)/2
sum(i=0...n-1) C(n,i)=sum(i=0...n) C(n,i) - 1=2^n - 1
f(x)=sum(i=0...n-1) C(n,i).i(i+1) x^(i) , x>0. une primitive de f est
g(x)=sum(i=0...n-1) C(n,i).i x^(i+1) à nouveau une primitive de g(x)/x² est
h(x)=sum(i=0...n-1) C(n,i).x^(i)= sum(i=0...n) C(n,i).x^(i)-x^n=(1+x)^n-x^n
==> g(x)=nx²(1+x)^(n-1) -n x^(n+1)
==> f(x)= 2nx(1+x)^(n-1)+n(n-1)x²(1+x)^(n-2) -n(n+1) x^(n)
==> sum(i=0...n-1) C(n,i).i(i+1) =f(1)=n2^(n)+n(n-1).2^(n-2) -n(n+1)
A= n(n+1)/2.(2^n - 1)-n2^(n-1)-n(n-1).2^(n-3) +n(n+1)/2
A= n(n+1)2^(n - 1)- n2^(n-1)-n(n-1).2^(n-3)
A= n(3n+1)2^(n-3) Sauf erreur
Accueil 
