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Question

Let Sk=1k+2k+...+nk and S(m,n)=mr=1m+1CrSr. Find S(3,2)

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Solution

Given Sk=1k+2k+....+nk .....(1)
Also given S(m,n)=mr=1m+1CrSr
S(3,n)=3r=14CrSr
S(3,n)=4C1S1+4C2S2+4C3S3
S(3,n)=4(1+2+....n)+6(12+22+....+n2)+4(13+23+....+n3)
S(3,n)=4n(n+1)2+6n(n+1)(2n+1)6+4n2(n+1)24
S(3,2)=12+30+36=78

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