The binomial coefficients appear as the entries of Pascal's triangle where each entry is the sum of the two above it.

One description of the types of simple-closed curves that may appear on the surface of the Klein bottle is given by the use of the first homology group of the Klein bottle calculated with integer coefficients.

Although this result might appear to settle the existence and uniqueness of solutions, there are examples of linear partial differential equations whose coefficients have derivatives of all orders ( which are nevertheless not analytic ) but which have no solutions at all: see Lewy ( 1957 ).

The required diffusion coefficients, however, appear to be larger than are observed in the real ocean.

That is, the cumulants appear as the coefficients in the Maclaurin series of g ( t ):

Just like " n choose k " are the coefficients when you raise a binomial to the n < sup > th </ sup > power ( e. g. the coefficients are 1, 3, 3, 1 for ( a + b )< sup > 3 </ sup >, where n = 3 ), the multinomial coefficients appear when one raises a multinomial to the n < sup > th </ sup > power ( e. g. ( a + b + c )< sup > 3 </ sup >)

The term " homophonic " is justified because the Dirichlet eigenvalues are precisely the fundamental tones that the drum is capable of producing: they appear naturally as Fourier coefficients in the solution wave equation with clamped boundary.

Static pressure and temperature appear as fixed coefficients defined by convention as standard sea level values.

Virial coefficients appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law.

It shows that the spherical multipole moments appear as coefficients in the 1 / R expansion of the potential.

Yang's writing represents the first in which quadratic equations with negative coefficients of ' x ' appear, although he attributes this to the earlier Liu Yi.

The coefficients of the characteristic will often appear very complicated, and the traces are not much better ; when looking for solutions it is almost always better to compute components of the Einstein tensor with respect to a suitably adapted frame and then to kill appropriate combinations of components directly.

The triangle demonstrates many mathematical properties in addition to showing binomial coefficients.

His Traité du triangle arithmétique (" Treatise on the Arithmetical Triangle ") of 1653 described a convenient tabular presentation for binomial coefficients, now called Pascal's triangle.

where are the binomial coefficients.

The binomial coefficients can be arranged to form Pascal's triangle.

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem.

Arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascal's triangle.

The properties of binomial coefficients have led to extending the meaning of the symbol beyond the basic case where n and k are nonnegative integers with ; such expressions are then still called binomial coefficients.

The earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, due to Halayudha, on an ancient Hindu classic, Pingala's chandaḥśāstra.

In about 1150, the Hindu mathematician Bhaskaracharya gave a very clear exposition of binomial coefficients in his book Lilavati.

This shows in particular that is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients ( counting problems for which the answer is given by a binomial coefficient expression ), for instance the number of words formed of n bits ( digits 0 or 1 ) whose sum is k is given by, while the number of ways to write where every a < sub > i </ sub > is a nonnegative integer is given by.

One has a recursive formula for binomial coefficients

This formula and the triangular arrangement of the binomial coefficients are often attributed to Blaise Pascal, who described them in the 17th century, but they were known to many mathematicians who preceded him.

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.

The binomial coefficients 1, 2, 1 appearing in this expansion correspond to the third row of Pascal's triangle.

# the nth row of the Pascal's Triangle will be the coefficients of the expanded binomial.

In quantum mechanics it often occurs that little or no information about the inner product of two arbitrary ( state ) kets is present, while it is possible to say something about the expansion coefficients and of those vectors with respect to a chosen ( orthonormalized ) basis.

* Thermal expansion coefficients of the elements ( data page ) — thermal expansion

They have fairly low coefficients of thermal expansion ( 7740 Pyrex CTE is 3. 25 /° C as compared to about 9 /° C for a typical soda-lime glass ), making them more dimensionally stable.

It has one of the lowest coefficients of thermal expansion among commercially used metals.

Even more limiting is the cracking of fuel coatings due to the large temperature ranges ( from 22 K up to 3000 K over the length of a 1. 3m fuel rod ), and the necessity of matching coefficients of expansion in all the components.

The integral representation for these types of series is interesting because the integral can often be evaluated using asymptotic expansion or saddle-point techniques ; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large n.

hence the above Newton interpolation formula ( by matching coefficients in the expansion of an arbitrary function f ( x ) in such symbols ), and so on.

These special transition joints have several short sections of special types of glass fused together between the metal and the usual type of glass, each having more gradual changes in thermal expansion coefficients.

The " weights ", or coefficients, of the terms in the Fourier expansion of a function can be thought of as components of a vector in an infinite dimensional space known as a Hilbert space.

Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients.

where the coefficients a < sub > i </ sub > in this expansion are precisely the numbers on row n of Pascal's triangle.

To see how the binomial theorem relates to the simple construction of Pascal's triangle, consider the problem of calculating the coefficients of the expansion of ( x + 1 )< sup > n + 1 </ sup > in terms of the corresponding coefficients of ( x + 1 )< sup > n </ sup > ( setting y

( a + b )< sup > n </ sup > = b < sup > n </ sup >( a / b + 1 )< sup > n </ sup >, the coefficients are identical in the expansion of the general case.

Stirling numbers of the first kind ( without the qualifying adjective unsigned ) are the coefficients in the expansion

The P < sub > n </ sub > can also be defined as the coefficients in a Taylor series expansion:

Replacing the quotient of the square root with its definition in ( 1 ), and equating the coefficients of powers of t in the resulting expansion gives Bonnet ’ s recursion formula

The total pyroelectric coefficient measured at constant stress is the sum of the pyroelectric coefficients at constant strain ( primary pyroelectric effect ) and the piezoelectric contribution from thermal expansion ( secondary pyroelectric effect ).

Antimony and tin are added to make the character produced durable and tough while reducing the difference between the coefficients of expansion of the matrix and the alloy.

we see that we have written the Fourier expansion of a function of the complex variable τ, so the coefficients of the q-series are also thought of as the Fourier coefficients of.