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Central Binomial Coefficient
In mathematics the ''n''th central binomial coefficient is the particular binomial coefficient : = \frac = \prod\limits_^\frac \textn \geq 0. They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first few central binomial coefficients starting at ''n'' = 0 are: :, , , , , , 924, 3432, 12870, 48620, ...; Properties The central binomial coefficients represent the number of combinations of a set where there are an equal number of two types of objects. For example, n=2 represents ''AABB, ABAB, ABBA, BAAB, BABA, BBAA''. They also represent the number of combinations of ''A'' and ''B'' where there are never more ''B'' 's than ''A'' 's. For example, n=2 represents ''AAAA, AAAB, AABA, AABB, ABAA, ABAB''. The number of factors of ''2'' in \binom is equal to the number of ones in the binary representation of ''n'', so ''1'' is the only odd central binomial coefficient. Generating function The ordinary generating fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal Triangle Small
Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, French mathematician, physicist, inventor, philosopher, writer and theologian Places * Pascal (crater), a lunar crater * Pascal Island (Antarctica) * Pascal Island (Western Australia) Science and technology * Pascal (unit), the SI unit of pressure * Pascal (programming language), a programming language developed by Niklaus Wirth * PASCAL (database), a bibliographic database maintained by the Institute of Scientific and Technical Information * Pascal (microarchitecture), codename for a microarchitecture developed by Nvidia Other uses * (1895–1911) * (1931–1942) * Pascal and Maximus, fictional characters in ''Tangled'' * Pascal blanc, a French white wine grape * Pascal College, secondary education school in Zaandam, the Netherlands * Pasca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling's Formula
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre. One way of stating the approximation involves the logarithm of the factorial: \ln(n!) = n\ln n - n +O(\ln n), where the big O notation means that, for all sufficiently large values of n, the difference between \ln(n!) and n\ln n-n will be at most proportional to the logarithm. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to use instead the binary logarithm, giving the equivalent form \log_2 (n!) = n\log_2 n - n\log_2 e +O(\log_2 n). The error term in either base can be expressed more precisely as \tfrac12\log(2\pi n)+O(\tfrac1n), corresponding to an approximate formula for the factorial itself, n! \sim \sqrt\left(\frac\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Of Bertrand's Postulate
In mathematics, Bertrand's postulate (actually a theorem) states that for each n \ge 2 there is a prime p such that n |
Squarefree Integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-free, but is not, because 18 is divisible by . The smallest positive square-free numbers are Square-free factorization Every positive integer n can be factored in a unique way as n=\prod_^k q_i^i, where the q_i different from one are square-free integers that are pairwise coprime. This is called the ''square-free factorization'' of . To construct the square-free factorization, let n=\prod_^h p_j^ be the prime factorization of n, where the p_j are distinct prime numbers. Then the factors of the square-free factorization are defined as q_i=\prod_p_j. An integer is square-free if and only if q_i=1 for all i > 1. An integer greater than one is the kth power of another integer if and only if k is a divisor of all i such that q_i\neq 1. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős Squarefree Conjecture
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-free, but is not, because 18 is divisible by . The smallest positive square-free numbers are Square-free factorization Every positive integer n can be factored in a unique way as n=\prod_^k q_i^i, where the q_i different from one are square-free integers that are pairwise coprime. This is called the ''square-free factorization'' of . To construct the square-free factorization, let n=\prod_^h p_j^ be the prime factorization of n, where the p_j are distinct prime numbers. Then the factors of the square-free factorization are defined as q_i=\prod_p_j. An integer is square-free if and only if q_i=1 for all i > 1. An integer greater than one is the kth power of another integer if and only if k is a divisor of all i such that q_i\neq 1. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolstenholme's Theorem
In mathematics, Wolstenholme's theorem states that for a prime number p \geq 5, the congruence : \equiv 1 \pmod holds, where the parentheses denote a binomial coefficient. For example, with ''p'' = 7, this says that 1716 is one more than a multiple of 343. The theorem was first proved by Joseph Wolstenholme in 1862. In 1819, Charles Babbage showed the same congruence modulo ''p''2, which holds for p \geq 3. An equivalent formulation is the congruence : \equiv \pmod for p \geq 5, which is due to Wilhelm Ljunggren (and, in the special case b = 1, to J. W. L. Glaisher) and is inspired by Lucas' theorem. No known composite numbers satisfy Wolstenholme's theorem and it is conjectured that there are none (see below). A prime that satisfies the congruence modulo ''p''4 is called a Wolstenholme prime (see below). As Wolstenholme himself established, his theorem can also be expressed as a pair of congruences for (generalized) harmonic numbers: :1+++\dots+ \equiv 0 \pmod \mbox :1+++\dot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gould's Sequence
Gould's sequence is an integer sequence named after Henry W. Gould that counts how many odd numbers are in each row of Pascal's triangle. It consists only of power of two, powers of two, and begins:. :1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, ... For instance, the sixth number in the sequence is 4, because there are four odd numbers in the sixth row of Pascal's triangle (the four bold numbers in the sequence 1, 5, 10, 10, 5, 1). Additional interpretations The th value in the sequence (starting from ) gives the highest power of 2 that divides the central binomial coefficient \tbinom, and it gives the numerator of 2^n/n! (expressed as a fraction in lowest terms). Gould's sequence also gives the number of live cells in the th generation of the Rule 90 cellular automaton starting from a single live cell.. It has a characteristic growing sawtooth wave, sawtooth shape that can be used to recognize physical processes that behave similarly to Rule 90.. Related sequences ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Of Two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative values, so there are 1, 2, and 2 multiplied by itself a certain number of times. The first ten powers of 2 for non-negative values of are: : 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100...000 or 0.00...001, just like a power of 10 in the decimal system. Computer science Two to the exponent of , written as , is the number of ways the bits in a binary word of length can be arranged. A word, interpreted as an unsigned integer, can represent values from 0 () to () inclusively. Corresponding signed integer values can be positive, negative and zero; see signed n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^(1-t)^\,dt for complex number inputs z_1, z_2 such that \Re(z_1), \Re(z_2)>0. The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol is a Greek capital beta. Properties The beta function is symmetric, meaning that \Beta(z_1,z_2) = \Beta(z_2,z_1) for all inputs z_1 and z_2.Davis (1972) 6.2.2 p.258 A key property of the beta function is its close relationship to the gamma function: : \Beta(z_1,z_2)=\frac. A proof is given below in . The beta function is also closely related to binomial coefficients. When (or , by symmetry) is a positive integer, it follows from the definition of the gamma function thatDavis (1972) 6.2.1 p.258 : \Beta(m,n) =\dfrac = \frac \B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M \ (z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Numbers
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894). The ''n''th Catalan number can be expressed directly in terms of binomial coefficients by :C_n = \frac = \frac = \prod\limits_^\frac \qquad\textn\ge 0. The first Catalan numbers for ''n'' = 0, 1, 2, 3, ... are :1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... . Properties An alternative expression for ''C''''n'' is :C_n = - for n\ge 0, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that ''C''''n'' is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. The Catalan numbers satisfy the recurrence relations :C_0 = 1 \quad \text \quad C_=\sum_^C_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wallis Product
In mathematics, the Wallis product for , published in 1656 by John Wallis, states that :\begin \frac & = \prod_^ \frac = \prod_^ \left(\frac \cdot \frac\right) \\ pt& = \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \; \cdots \\ \end Proof using integration Wallis derived this infinite product as it is done in calculus books today, by examining \int_0^\pi \sin^n x\,dx for even and odd values of n, and noting that for large n, increasing n by 1 results in a change that becomes ever smaller as n increases. Let :I(n) = \int_0^\pi \sin^n x\,dx. (This is a form of Wallis' integrals.) Integrate by parts: :\begin u &= \sin^x \\ \Rightarrow du &= (n-1) \sin^x \cos x\,dx \\ dv &= \sin x\,dx \\ \Rightarrow v &= -\cos x \end :\begin \Rightarrow I(n) &= \int_0^\pi \sin^n x\,dx \\ pt &= -\sin^x\cos x \Biggl, _0^\pi - \int_0^\pi (-\cos x)(n-1) \sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
