Translation for "binomikertoimen" to english

Binomikertoimen

• binomial coefficient

Translation examples

binomial coefficient

Pascalin kolmio on matematiikassa binomikertoimista kolmion muotoon koottu järjestelmä.
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients.

Käytettäessä summamerkintää se voidaan kirjoittaa ( x + y ) 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 \choose k}x^{k}y^{n-k}.} Viimeinen lauseke seuraa edellisestä ja on symmetrinen x :n ja y :n ensimmäisen lausekkeen kanssa, ja verrattaessa kertoimiin huomataan, että binomikertoimien jono kaavassa on myös symmetrinen.
Using summation notation, it can be written as ( x + y ) 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 \choose k}x^{k}y^{n-k}.} The final expression follows from the previous one by the symmetry of x and y in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical.

Tämä lasketaan binomikertoimen ( N 2 ) {\displaystyle {\tbinom {N}{2}}} avulla, josta saadaan tiheyden kaava D = 2 ( E − N + 1 ) N ( N − 3 ) + 2 . {\displaystyle D={\frac {2(E-N+1)}{N(N-3)+2}}.} Suuntaamattoman verkoston tapauksessa kaava muuntuu muotoon D = T − 2 N + 2 N ( N − 3 ) + 2 , {\displaystyle D={\frac {T-2N+2}{N(N-3)+2}},} (Wasserman & Faust 1994).
The density D {\displaystyle D} of a network is defined as a ratio of the number of edges E {\displaystyle E} to the number of possible edges in a network with N {\displaystyle N} nodes, given (in the case of simple graphs) by the binomial coefficient ( N 2 ) {\displaystyle {\tbinom {N}{2}}} , giving D = E − ( N − 1 ) E m a x − ( N − 1 ) = 2 ( E − N + 1 ) N ( N − 3 ) + 2 {\displaystyle D={\frac {E-(N-1)}{Emax-(N-1)}}={\frac {2(E-N+1)}{N(N-3)+2}}} Another possible equation is D = T − 2 N + 2 N ( N − 3 ) + 2 , {\displaystyle D={\frac {T-2N+2}{N(N-3)+2}},} whereas the ties T {\displaystyle T} are unidirectional (Wasserman & Faust 1994).

Luvut ovat nimetty belgialaisen matemaatikon Eugène Charles Catalanin mukaan. n:s Catalanin luku lasketaan binomikertoimilla seuraavasti: C n = 1 n + 1 ( 2 n n ) = ( 2 n ) ! ( n + 1 ) ! n ! kun n ≥ 0. {\displaystyle C_{n}={\frac {1}{n+1}}{2n \choose n}={\frac {(2n)!}{(n+1)!\,n!}}\qquad {\mbox{ kun }}n\geq 0.} Ensimmäiset Catalanin luvut ovat (n arvoilla 0, 1, 2..): 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, … Tämä matematiikkaan liittyvä artikkeli on tynkä.
The nth Catalan number is given directly in terms of binomial coefficients by C n = 1 n + 1 ( 2 n n ) = ( 2 n ) ! ( n + 1 ) ! n ! = ∏ k = 2 n n + k k for n ≥ 0. {\displaystyle C_{n}={\frac {1}{n+1}}{2n \choose n}={\frac {(2n)!}{(n+1)!\,n!}}=\prod \limits _{k=2}^{n}{\frac {n+k}{k}}\qquad {\text{for }}n\geq 0.} The first Catalan numbers for n = 0, 1, 2, 3, ... are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, ... (sequence A000108 in the OEIS).

Lauseen mukaan on mahdollista kehittää mikä tahansa potenssi (x + y):n summaksi, joka on muotoa ( x + y ) n = ( n 0 ) x n y 0 + ( n 1 ) x n − 1 y 1 + ( n 2 ) x n − 2 y 2 + ⋯ + ( n n − 1 ) x 1 y n − 1 + ( n n ) x 0 y n , {\displaystyle (x+y)^{n}={n \choose 0}x^{n}y^{0}+{n \choose 1}x^{n-1}y^{1}+{n \choose 2}x^{n-2}y^{2}+\cdots +{n \choose n-1}x^{1}y^{n-1}+{n \choose n}x^{0}y^{n},} missä jokainen ( n k ) {\displaystyle {\tbinom {n}{k}}} on tietty positiivinen kokonaisluku, joka tunnetaan binomikertoimena.
According to the theorem, it is possible to expand any power of x + y into a sum of the form ( x + y ) n = ( n 0 ) x n y 0 + ( n 1 ) x n − 1 y 1 + ( n 2 ) x n − 2 y 2 + ⋯ + ( n n − 1 ) x 1 y n − 1 + ( n n ) x 0 y n , {\displaystyle (x+y)^{n}={n \choose 0}x^{n}y^{0}+{n \choose 1}x^{n-1}y^{1}+{n \choose 2}x^{n-2}y^{2}+\cdots +{n \choose n-1}x^{1}y^{n-1}+{n \choose n}x^{0}y^{n},} where each ( n k ) {\displaystyle {\tbinom {n}{k}}} is a specific positive integer known as a binomial coefficient.