南京大学：《组合数学 Combinatorics》课程教学资源（课件讲稿）Catalan Number

Changing Money 件 50 100 (壹，伍，拾，贰拾，伍拾，壹佰)nxn m≥0 1 (1-x)(1-x5)(1-x10)(1-x20)(1-2x50)(1-x100)

Changing Money ￾ n￾0 (၊ ໿ ൎ م ൎ໿ ൎ၊ϭ)nxn = 1 (1 ￾ x)(1 ￾ x5)(1 ￾ x10)(1 ￾ x20)(1 ￾ x50)(1 ￾ x100)

>(壹，伍，拾，贰拾，伍拾，壹佰)mxn m>0 1 (1-x)(1-x5)(1-x10)(1-x20)(1-x50)(1-x100) =(1+x+x2+x3++x99)-(1+x5+x10+x15+·+95) -(1+x10+x20+x30+…+x90)-(1+x20+x40+x60+x80) -(1+x50) (1-2x100)6 Newton's formula ∑(n9)(

￾ n￾0 (၊ ໿ ൎ م ൎ໿ ൎ၊ϭ)nxn = 1 (1 ￾ x)(1 ￾ x5)(1 ￾ x10)(1 ￾ x20)(1 ￾ x50)(1 ￾ x100) = · 1 (1 ￾ x100)6 (1 + x + x2 + x3 + ··· + x99)·(1 + x5 + x10 + x15 + ··· + x95) ·(1 + x10 + x20 + x30 + ··· + x90)·(1 + x20 + x40 + x60 + x80) ·(1 + x50) = Newton’s formula ￾ n￾0 ￾￾6 n ￾ ￾ ￾x100￾n

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