çµåãé¢æ°ã§æç«ãã代衚çãªãã®ãšããŠ
nC0+nC1+nC2++nCn=2^n
ãããã
ãã®ãµã€ãã§ãä»ã®ãããããªçåŒãæç«ããã³ãŒããŒã確ããã£ããããª
å°è±¡ãããæ¢ããèšå€§ãªå
容ãå«ãã§ããã®ã§äœåŠã ã£ããèŠã€ãããããªã®ã§
ããã«çŽ¹ä»ãããŠããããç¥ããŸããããè²ã
ãšäŸãäžããŠã¿ãŸãã®ã§ææŠããŠ
èŠãŠäžããã
nCkãå«ãŸãããšãç§ã¯çŽæã§ã¯ãªããªãæ°ãä»ããŸããã
(1)nãå¶æ°ã®ãšã
nC0+nC2+nC4++nCn
(2)nãå¥æ°ã®ãšã
nC0+nC2+nC4++nC[n-1]
(3)kã®æ¹ãåºå®ãã
nCk+[n+1]Ck+[n+2]Ck++[n+m]Ck
(4)n,kãåæã«å€åããã
nCk+[n-1]C[k-1]+[n-2]C[k-2]++[n-k]C0
(5)n,k,笊å·ãåæã«å€åããã
2nC0-[2n-1]C1+[2n-2]C2-[2n-3]C3++(-1)^n*nCn
(6)2ã€ã®Cé¢æ°ã®ç©ãçµåãã
nC0*mCk+nC1*mC[k-1]+nC2*mC[k-2]++nCk*mC0
(7)2ã€ã®Cé¢æ°ã®ç©ã笊å·ã亀äºã«çµåãã
nC0*nCk-nC1*[n-1]S[k-1]+nC2*[n-2]C[k-2]-+(-1)^k*nCk*[n-k]C0
(8)Cã«ä¿æ°ãä»éããã
2nCn+2*[2n-1]Cn+2^2*[2n-2]Cn++2^n*nCn
http://shochandas.xsrv.jp/number/binomialcoefficient.htm
âãã®ããŒãžã§ããã
ãããŠ
(1)ãš(2)ã¯äžèšããŒãžã®(6)
(3)ã¯äžèšããŒãžã®(11)
(6)ã¯äžèšããŒãžã®(16)
ã«çžåœããŸããã
(4)(5)(7)(8)ã¯ãªãããã§ããããæ¢ãæ¹ãæªãã ãããç¥ããŸããã
http://shochandas.xsrv.jp/number/binomialcoefficient.htm
ã®ããŒãžãèŠãŠããŸããã
(5)ã¯äžèšããŒãžã®(12)
(8)ã¯äžèšããŒãžã®(13)
ã«çžåœããŸããã
(4)ã¯ãäžèšããŒãžã®(11)ã®èª¬ææã®ãªãã®
ã(11)ã§ãk=nã®ãšãã¯ãâŠâŠãç·åã®å
¬åŒããšãèšããããã
ã®éšåã«æžãããŠããåŒã«ãããŠnã«n-kã代å
¥ããŠmã«kã代å
¥ãããã®ãªã®ã§ãçãã¯[n+1]Ck
(7)ã¯ãäžèšããŒãžã®(27)ã«ãããŠäž¡èŸºã«(-1)^kããããŠnã«n-kã代å
¥ããŠmã«nã代å
¥ãããã®ãªã®ã§ãçãã¯(-1)^k*[k-1]Ck=0
ããïœå
šéšæ¢ã«ã¢ãããããŠãããã ã
ããã«ãªããã®ãäœã£ãŠã¿ãŸããã
(1)aCb*cC0+[a+1]Cb*cC1+[a+2]Cb*cC2+[a+3]Cb*cC3++[a+k]Cb*cCk++[a+c]Cb*cCc
(ãã ãaâ§bâ§câ§0)
(2) aC0*sCa - aC1*[s-t]Ca + aC2*[s-2*t]Ca - aC3*[s-3*t]Ca + +(-1)^k*aCk*[s-k*t]Ca++(-1)^a*aCa*[s-a*t]Ca
(ãã ãa,s,t>0ã®æŽæ°)