数列の和
Σ_{k=1～n}k=n(n+1)/2=C(n+1,2)
Σ_{k=1～n}k(k+1)=n(n+1)(n+2)/3=2C(n+2,3)
Σ_{k=1～n}k(k+1)(k+2)=(1/4)Σ_{k=1～n}k(k+1)(k+2)(k+3)-(k-1)k(k+1)(k+2)
=n(n+1)(n+2)(n+3)/4=3!C(n+3,4)
Σ_{k=1～n}k(k+1)(k+2)(k+3)(k+4)
=(1/5)Σ_{k=1～n}k(k+1)(k+2)(k+3)(k+4)-(k-1)k(k+1)(k+2)(k+3)
=n(n+1)(n+2)(n+3)(n+4)/5=4!C(n+4,5)
...
Σ_{k=1～n}k(k+1)(k+2)(k+3)(k+4)...(k+m-1)=(m-1)!C(n+m-1,m)
と、
C(n,k)=C(n-1,k)+C(n-1,k-1)
C(n-1,k-1)=C(n,k)-C(n-1,k)
C(n,k)=C(n+1,k+1)-C(n,k+1)
=C(n+2,k+2)-2C(n+1,k+2)+C(n,k+2)
=C(n+3,k+3)-3C(n+2,k+3)+3C(n+1,k+3)-C(n,k+3)
=Σ_{j=0～m}(-1)^j*C(m,j)*C(n+m-j,k+m)
より、
Σ_{k=1～n}k^2=Σ_{k=1～n}k(k+1)-Σ_{k=1～n}k
=2C(n+2,3)-C(n+1,2)=2C(n+2,3)-(C(n+2,3)-C(n+1,3))=C(n+2,3)+C(n+1,3)
Σ_{k=1～n}k^3=Σ_{k=1～n}k(k+1)(k+2)-3Σ_{k=1～n}k(k+1)+Σ_{k=1～n}k
=3!C(n+3,4)-3*2C(n+2,3)+C(n+1,2)
=6C(n+3,4)-6C(n+2,3)+C(n+1,2)
=6C(n+3,4)-6(C(n+3,4)-C(n+2,4))+C(n+3,4)-2C(n+2,4)+C(n+1,4)
=C(n+3,4)+4C(n+2,4)+C(n+1,4)
Σ_{k=1～n}k^4
=Σ_{k=1～n}k(k+1)(k+2)(k+3)-6Σ_{k=1～n}k(k+1)(k+2)
+7Σ_{k=1～n}k(k+1)-Σ_{k=1～n}k
=4!C(n+4,5)-6*3!C(n+3,4)+7*2C(n+2,3)-C(n+1,2)
=24C(n+4,5)-36C(n+3,4)+14C(n+2,3)-C(n+1,2)
=24C(n+4,5)-36(C(n+4,5)-C(n+3,5))+14(C(n+4,5)-2C(n+3,5)+C(n+2,5))-(C(n+4,5)-3C(n+3,5)+3C(n+2,5)-C(n+1,5))
=C(n+4,5)+11C(n+3,5)+11C(n+2,5)+C(n+1,5)
Σ_{k=1～n}k^5
=Σ_{k=1～n}k(k+1)(k+2)(k+3)(k+4)-10Σ_{k=1～n}k(k+1)(k+2)(k+3)
+25Σ_{k=1～n}k(k+1)(k+2)-15Σ_{k=1～n}k(k+1)+Σ_{k=1～n}k
=5!C(n+5,6)-10*4!C(n+4,5)+25*3!C(n+3,4)-15*2C(n+2,3)+C(n+1,2)
=120C(n+5,6)-240C(n+4,5)+150C(n+3,4)-30C(n+2,3)+C(n+1,2)
=120C(n+5,6)-240(C(n+5,6)-C(n+4,6))+150(C(n+5,6)-2C(n+4,6)+C(n+3,6))
-30(C(n+5,6)-3C(n+4,6)+3C(n+3,6)-C(n+2,6))+(C(n+5,6)-4C(n+4,6)+6C(n+3,6)-4C(n+2,6)+C(n+1,6))
=C(n+5,6)+26C(n+4,6)+66C(n+3,6)+26C(n+2,6)+C(n+1,6)
Σ_{k=1～n}k^6
=Σ_{k=1～n}k(k+1)(k+2)(k+3)(k+4)(k+5)-15Σ_{k=1～n}k(k+1)(k+2)(k+3)(k+4)
+65Σ_{k=1～n}k(k+1)(k+2)(k+3)-90*Σ_{k=1～n}k(k+1)(k+2)
+31*Σ_{k=1～n}k(k+1)-Σ_{k=1～n}k
=6!C(n+6,7)-15*5!C(n+5,6)+65*4!C(n+4,5)-90*3!C(n+3,4)+31*2C(n+2,3)-C(n+1,2)
=720C(n+6,7)-1800C(n+5,6)+1560C(n+4,5)-540C(n+3,4)+62C(n+2,3)-C(n+1,2)
=C(n+6,7)+57C(n+5,7)+302C(n+4,7)+302C(n+3,7)+57C(n+2,7)+C(n+1,7)
比べてみたらとても興味深かったです。↓↓↓
https://www.chart.co.jp/subject/sugaku/suken_tsushin/108/108-7.pdf
