S[n]==((∑_(k=1)^n)S[k-1]+S[n-k]+1), S[0]==0,S[1]==1
S[n]==2(∑_(k=1)^n)S[k-1]+n, S[0]==0,S[1]==1
S[n]==2(∑_(k=0)^(n-1))S[k]+n, S[0]==0,S[1]==1
S[n+1]==2(∑_(k=0)^n)S[k]+n+1, S[0]==0,S[1]==1

RSolve[{S[n+1]==n+2(∑_(i=0)^n)S[i], S[0]==0,S[1]==1},S[n], n]

S[n+1]-S[n]=2S[n]+1
S[n+1]-3S[n]=1

RSolve[{S[n+1]-3S[n]==1, S[0]==0,S[1]==1},S[n], n]