C0C1+2C1C2+3C2C3........+nC(n-1)Cn find the sum of this bino-binomial series.

LHS=(C0​+C1​)(C1​+C2​)(C2​+C3​)....(Cn−1​+Cn​)
=∏r=1n​(Cr−1​+Cr​)
=∏r=1n​n+1Cr​(∵nCr​+nCr−1​=n+1Cr​)
=∏r=1n​(rn+1​)nCr−1​(∵nCr​=rn​n−1Cr−1​)
=∏nr=1​(n+1)∏nr=1​1/r∏r=1n​nCr−1​
=(n+1)nn!1​(C0​C1​....Cn−1​)
=n!(n+1)n​(C0​C1​.......Cn−1​)=RHS