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Ⅰ.組合数級数与它的和由组合公式: C_x~r=x(x-1)(x-2)…(x-r+1)/r!, C_(ax+b)~r=(ax+b)(ax+b-1)(ax+b-2)…(ax+b-r+1)/r!, C_(a_0x~s+a_1x~(s-1)+…+a_s)~r=(a_0x~s+a_1x~(s-1)+…+a_s)..(a_0x~s+a_1x~(s-1)+…+ a_s-1)(a_0x~s++a_1x~(s-1)+…+a_s-2)… ..(a_0x~s+a_1x~(s-1)+…+a_s-r+1)/r!, 可知C_x~r,C_(ax+b)~r为x的r次函数,C_(a_0x~s+a_1x~(s-1)+…+a_s)~r为x的rs次函数。因此当x取連續整数时,C_x~r,c_(ax+b)~r的数列是r阶等差級数;C_(a_0x~s+a_1x~(s-1)+…+as)~r的数列是rs阶等差級数。或者說:从連續整数或等差級数(x取連續整数时ax+b的数列是等差級数)中取r的組合数的数列是r阶等差級数;从s阶等差級数(x取連續整数时a_0x~s+a_1x~(s-1)+…+a_s的数列是s阶等差級数)中取r的組合数的数列是rs阶等差級数。
I. The combination of the number of levels and its sum by the formula: C_x~r=x(x-1)(x-2)...(x-r+1)/r!, C_(ax+b)~r=( Ax+b)(ax+b-1)(ax+b-2)...(ax+b-r+1)/r!, C_(a_0x~s+a_1x~(s-1)+...+a_s) ~r=(a_0x~s+a_1x~(s-1)+...+a_s)..(a_0x~s+a_1x~(s-1)+...+a_s-1)(a_0x~s++a_1x~(a_0x~s++a_1x~( S-1)+...+a_s-2)... ..(a_0x~s+a_1x~(s-1)+...+a_s-r+1)/r!, know C_x~r, C_(ax+b) ~r is an r-th function of x, and C_(a_0x~s+a_1x~(s-1)+...+a_s)~r is an rs subfunction of x. Therefore, when x is a continuous integer, the sequence of C_x~r,c_(ax+b)~r is an r-order series of equal differences; C_(a_0x~s+a_1x~(s-1)+...+as)~r The series is equal to the rs order. Or say: From a continuous integer or equal-order series (x is a continuous integer when the series ax + b is equal to the series) is the number of combinations of r is equal to the r-order series; from the s-order series (x The series of a_0x~s+a_1x~(s-1)+...+a_s is a series of s-order equal-difference series).