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Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written ( n k ) . {\displaystyle {\tbinom {n}{k}}.} It is the coefficient...

) = ∑ k = 0 ∞ F k z k − ∑ k = 0 ∞ F k z k + 1 − ∑ k = 0 ∞ F k z k + 2 = ∑ k = 0 ∞ F k z k − ∑ k = 1 ∞ F k − 1 z k − ∑ k = 2 ∞ F k − 2 z k = 0 z 0 + 1...

n = ∑ k = 0 n ( n k ) x n − k y k = ∑ k = 0 n ( n k ) x k y n − k . {\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}=\sum _{k=0}^{n}{n...

Poisson distribution. What is the probability of k = 0 meteorite hits in the next 100 years? P ( k = 0 meteorites hit in next 100 years ) = 1 0 e − 1 0...

∏ k = 0 n f k − f n a n ∏ k = 0 n f k = g n ∏ k = 0 n f k {\displaystyle {\frac {a_{n+1}}{\prod _{k=0}^{n}f_{k}}}-{\frac {f_{n}a_{n}}{\prod _{k=0}^{n}f_{k}}}={\frac...

series gives ∑ k = 0 ∞ sin ⁡ ( k x ) r k = 1 2 i [ ∑ k = 0 ∞ ( e i x r ) k − ∑ k = 0 ∞ ( e − i x r ) k ] . {\displaystyle \sum _{k=0}^{\infty }{\frac...

as follows: s n = ∏ k = 0 n ( n k ) = ∏ k = 0 n n ! k ! ( n − k ) ! {\displaystyle s_{n}=\prod _{k=0}^{n}{n \choose k}=\prod _{k=0}^{n}{\frac {n!}{k!(n-k)...

j ) ] = ∑ k = 0 j − 1 1 ( n − k ) λ E ⁡ [ X ( i ) ] + E ⁡ [ X ( i ) 2 ] = ∑ k = 0 j − 1 1 ( n − k ) λ ∑ k = 0 i − 1 1 ( n − k ) λ + ∑ k = 0 i − 1 1 (...

equation through by ⟨ k ( 0 ) | {\displaystyle \langle k^{(0)}|} gives ( E n ( 0 ) − E k ( 0 ) ) ⟨ k ( 0 ) | n ( 1 ) ⟩ = ⟨ k ( 0 ) | V | n ( 0 ) ⟩ . {\displaystyle...

= { p if  k = 1 , q = 1 − p if  k = 0. {\displaystyle f(k;p)={\begin{cases}p&{\text{if }}k=1,\\q=1-p&{\text{if }}k=0.\end{cases}}} This can also be expressed...

F k 0 ( x ) = { 1 , if  x ≥ k 0 0 , if  x < k 0 {\displaystyle F_{k_{0}}(x)=\left\{{\begin{matrix}1,&{\mbox{if }}x\geq k_{0}\\0,&{\mbox{if }}x<k_{0}\end{matrix}}\right...

Swain–Scott equation derived in 1953: log 10 ⁡ ( k k 0 ) = s n {\displaystyle \log _{10}\left({\frac {k}{k_{0}}}\right)=sn} This free-energy relationship relates...

∏ k = 0 n − 1 ( a 1 + k d ) = ∏ k = 0 n − 1 d ( a 1 d + k ) = d ( a 1 d ) d ( a 1 d + 1 ) d ( a 1 d + 2 ) ⋯ d ( a 1 d + ( n − 1 ) ) = d n ∏ k = 0 n −...

following power series: exp ⁡ x := ∑ k = 0 ∞ x k k ! = 1 + x + x 2 2 + x 3 6 + x 4 24 + ⋯ {\displaystyle \exp x:=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}=1+x+{\frac...

Grothendieck-group K 0 ( X ) {\displaystyle K_{0}(X)} which is isomorphic to K 0 ( X ) {\displaystyle K^{0}(X)} if X {\displaystyle X} is smooth. The group K 0 ( X )...

parameter gives the coefficient of variation: σ / μ = k − 0.5 = 1 / α {\displaystyle \sigma /\mu =k^{-0.5}=1/{\sqrt {\alpha }}} The skewness of the gamma...

Let n ≥ 1. Then ∑ k = 0 n ( n + 1 k ) ( n + k + 1 ) B n + k = 0 {\displaystyle \sum _{k=0}^{n}{\binom {n+1}{k}}(n+k+1)B_{n+k}=0} Let n ≥ 0. Then (Leopold...

The Natal Government Railways Class K 0-6-0ST of 1880 was a South African steam locomotive from the pre-Union era in the Colony of Natal. In 1880, the...

Y ) = ∑ k = 0 ∞ ( 1 − p ) k p ⋅ k = p ∑ k = 0 ∞ ( 1 − p ) k k = p ( 1 − p ) ∑ k = 0 ∞ ( 1 − p ) k − 1 ⋅ k = p ( 1 − p ) [ d d p ( − ∑ k = 0 ∞ ( 1 − p...

⋅ δ r k = 0 , {\displaystyle \sum _{k=1}^{N}\mathbf {C} _{k}\cdot \delta \mathbf {r} _{k}=0\,,} so that ∑ k = 1 N ( N k − m k a k ) ⋅ δ r k = 0 . {\displaystyle...