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杨辉三角形, 二项式定理与多项式定理

[二项式定理] \begin{array}{l} \left( {a + b} \right)^n = C_n^0 a^n + C_n^1 a^{n - 1} b + C_n^2 a^{n - 2} b^2 + \cdots + C_n^{n - 1} ab^{n - 1} + C_n^n b^n \\ = \sum\limits_{k = 0}^n {C_n^k a^{n - k} b^k } \\ \end{array} 式中n为正整数, C_n^k 称为二项系数.

[杨辉三角形] 我国南宋时期数学家杨辉在他所著的《详解九章算法》(1261年)中记载着有关二项系数的研究.在二项式定理中,当n分别取0, 1, 2, 3, 4, 5, 6时,其二项系数表示成图

即所谓“杨辉三角形”.法国人帕斯卡也有类似结果(1650年),故外国书刊中称之为“帕斯卡三角形”,但比杨辉晚了近四百年.

[多项式定理]  \left( {a + b + \cdots + f} \right)^n = \sum\limits_{p + q + \cdots + s = n}^{} {\frac{{n!}}{{p!q! \cdots s!}}a^p b^q \cdots f^s } 和式中每一数组(p,q,…,s)对应一项,这个数组满足0≤pn,0≤qn,…,0≤snp+q+…+s=n,Σ是对于所有这样的数组求和.



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参阅
  1. 数学 - 数学符号 - 数学索引
  2. 手册 = 初中数学手册 + 高中数学手册 + 数学手册 + 实用数学手册
  3. 初等数学 = 小学数学 + 中学数学 ( 初中数学 + 高中数学 )
  4. 高等数学 = 基础数学 ( 代数 + 几何 + 分析 ) + 应用数学
  5. 公式 - 定理 - - 函数图 - 曲线图 - 平面图 - 立体图 - 动画 - 画画
  6. 书单 = 数学 + 物理 + 化学 + 计算 + 医学 + 英语 + 教材 - QQ群下载书
  7. 数学手册计算器 = 数学 + 手册 + 计算器 + 计算机代数系统
  8. 检测 - 例题 :

(d^0.5y)/dx^0.5 = sin(x-1)*sin(y-1) == ? (d^0.5y)/dx^0.5 -cosh(y)-sinh(y)=0 == ? (d^1.6y)/(dx^1.6)-int y(x) (dx)^(0.8)-y-exp(x)=0 == ? int y(x) (dx)^0.5 -y-exp(x)=0 == ? (d^0.5y)/dx^0.5-exp(y)*x=0 == ? (d^0.5y)/dx^0.5-exp(y)*y=0 == ? (d^0.5y)/dx^0.5=cos(x)/x*y == ? y*(dy^0.5)/dx^0.5-sqrt(x)-1=0 == ? (d^1.2y)/(dx^1.2)-2(d^0.6y)/dx^0.6+y-exp(x)=0 == ? (d^0.5y)/dx^0.5=cos(y)*exp(x)*x == ? (d^1.6y)/(dx^1.6)-2(d^0.8y)/dx^0.8+y-exp(x)=0 == ? (d^0.5y)/dx^0.5-exp(y)*sqrt(x)=0 == ? (d^1.6y)/(dx^1.6)-3 (d^0.8y)/dx^0.8+2y-exp(x)=0 == ? (d^0.5y)/dx^0.5 +log(y-1)-exp(x)-x=0 == ? (d^0.5y)/dx^0.5-exp(y)*sin(x)=0 == ? (d^0.5y)/dx^0.5 = y*sin(x)/x == ? y^((0.5))(x) -4 exp(x)*y-exp(x)=0 == ? (dy^0.5)/dx^0.5 = 1/(x-y) == ? dy/dx-(d^0.5y)/dx^0.5 - y - exp(x)=0 == ? (dy)/dx -exp(y-1)-x-x^2=0 == ? (d^1.2y)/(dx^1.2)-3dy^0.6/dx^0.6+2y-exp(x)=0 == ? dy/dx-(d^0.5y)/dx^0.5-y-1=0 == ? (d^0.5y)/dx^0.5-cos(y)*sin(x)=0 == ? (d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(4x)=0 == ? dy/dx-exp(y-1)-exp(x)=0 == ? (dy)/dx - 2(d^0.5y)/dx^0.5-y-exp(x)=0 == ? (d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(x)=0 == ? (d^0.5y)/dx^0.5 -e^(4x)-y=0 == ? y^((0.5))(x) - exp(x)*y-exp(x)=0 == ? y^((0.5))(x) - exp(x)*y-4exp(x)=0 == ? (dy)/dx -3(d^0.5y)/dx^0.5 +2y-exp(x)=0 == ? y*(d^0.5y)/dx^0.5-sqrt(x)-1=0 == ? y^((1))(x)-exp(y-1)-x=0 == ? (d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-2y-exp(x)=0 == ? (d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(4x)=0 == ? (d^0.5y)/dx^0.5 - log(y-1) - exp(x) + x=0 == ? (dy)/dx +asin(y-1) - cos(x)-x=0 == ? (d^1.6y)/(dx^1.6)-3(d^0.8y)/dx^0.8+2y-exp(x)=0 == ? (dy)/(dx) -sqrt(y-1)-x-1 =0 == ? (dy)/(dx) -exp(y-1)-exp(x) = 0 == ? (dy)/dx +asinh(y-1)-cosh(x)-x =0 == ? ((d^(1/2)y)/dx^(1/2))^2 -3y* (dy^0.5)/dx^0.5 + 2y^2 = 0 == ? (dy^0.5)/dx^0.5 = cos(x)*cos(y-1) == ? (d^0.5y)/dx^0.5 +log(y-1)-exp(x)-x=0 == ? (dy^0.5)/dx^0.5 = sin(x-1)*exp(y-1) == ? y*(d^2y)/dx^2-(dy/dx)^2+1=0 == ? y^((1))(x)-exp(y-1)-log(x)=0 == ? (d^2y)/dx^2 *exp(x)- exp(y-1)=0 == ? (d^1.6y)/(dx^1.6)-2 (d^0.8y)/dx^0.8-y-exp(x)=0 == ? (d^1.6y)/(dx^1.6)-2 (d^0.8y)/dx^0.8+y-exp(x)=0 == ? (dy)/dx -3 (d^0.5y)/dx^0.5+2y-exp(x)=0 == ? y^((0.5))(x) - x*y-x=0 == ? y*(dy^3)/dx^3-x^3-3x^2-3x-1=0 == ? y^((1.8))(x)-2y^((0.9))(x) +y-1=0 == ? y^((0.5))(x)=1/(x*y-1) == ? y^((2))(x)*y^2-x^2-2x-1=0 == ? ((d^0.5y)/dx^0.5)^2 -5(d^0.5y)/dx^0.5 +6=0 == ? y^((0.5))(x) -2 exp(x)*y-4exp(x)=0 == ? (d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(x)=0 == ? y^(0.5)(x)=2y*exp(x) == ? y^((0.5))(x)-exp(x)*y^2=0 == ? (d^1.6y)/(dx^1.6)-2(d^0.8y)/dx^0.8+y-exp(x)=0 == ? y^((1))(x)-y^2-x*y=0 == ? y^((1))(x)-y^((0.5))(x) -y-1=0 == ? y^((2))(x) -y^2-x^2=0 == ? y^((2))(x) -y^2-x^2-2x*y=0 == ? y^((0.5))(x) -int y(x) (dx)^0.5-y-exp(x)=0 == ? d^0.5/dx^0.5 y -2cos(y)*exp(x)=0 == ? d^0.5/dx^0.5 y -4sin(y)*exp(x)=0 == ? (d^0.5y)/dx^0.5=sin(x^2)*y == ? (d^0.5y)/dx^0.5-sin(x)*sin(y)=0 == ? (d^0.5y)/dx^0.5-sinh(x)*sinh(y)=0 == ? y^((1))(x)=exp(x-y)-x == ? x*(d^0.5y)/dx^0.5-y-2x=0 == ? (d^0.5y)/dx^0.5=sinh(x-1)*sinh(y-1) == ? y^((0.5))(x)-exp(-x)*y^2=0 == ? (d^0.5y)/dx^0.5=y/x*sin(x) == ? (dy)/dx-sin(x-y)-1=0 == ? (d^2.5y)/dx^2.5=y*(d^0.5y)/dx^0.5 == ? (d^0.5y)/dx^0.5=y*(dy)/dx == ? (d^(2-i)y)/dx^(2-i)- y+x=0 == ? (d^2y)/dx^2=y^3*x^2 == ? y*(d^2y)/dx^2-x^2-3x-1=0 == ? y*(d^2y)/dx^2-2x^2-3x-1=0 == ? (y-x-1)*(d^2y)/dx^2-3x-1=0 == ? y^2*(d^2y)/dx^2-x^2-4x-4=0 == ? (y-x-1)*(d^2y)/dx^2-x^2-4x-4=0 == ? y*(d^2y)/dx^2-2x^2-2x-1=0 == ? y*(d^3y)/dx^3-6x^3-3x^2-3x-1=0 == ? y^((0))(x)*y^((1))(x)*y^((2))(x)=x^2 == ? y^((3))(x)*y^((2))(x)=y^((1/2))(x) == ? y^((3))(x)=exp(x)*y^((1))(x)*y^((1/2))(x) == ? y^((1/2))(x)*y^((3))(x)=exp(x) == ? y^((1/2))(x)*y^((2))(x)=exp(x) == ? (d^0.5y)/dx^0.5-2x*y-1=0 == ? y^2*(d^0.5y)/dx^0.5-x^2-4x-4=0 == ? exp(y-1)*(d^0.5y)/dx^0.5-x=0 == ? y*(d^2y)/dx^2-(x-2)*(2x-4)=0 == ? y*(d^3y)/dx^3-6x^3-4x^2-4x-1=0 == ? exp(y-1)*(d^2y)/dx^2-exp(x)=0 == ? y^2*(d^2y)/dx^2-x^2-1=0 == ? 1/y^2*(d^2y)/dx^2-x^2-1=0 == ? (y-x-1)*(d^3y)/dx^3-(x-2)*(2x-4)*(3x-1)=0 == ? (d^0.5y)/dx^0.5-2x^2*y^2-8x^2=0 == ? (d^0.5y)/dx^0.5-2x*y^2-8x=0 == ? (d^0.5y)/dx^0.5-y^2-2y-2=0 == ? (d^0.5y)/dx^0.5-log(y-1)*exp(x)=0 == ? y*(d^2y)/dx^2-(dy/dx)^2-1=0 == ? (d^2y)/dx^2-asin(y-1)-sin(x)-x=0 == ? dy/dx*(x--y)-x--y-1 = 0 == ?


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