Pascal’s triangle, table

The relative intensities of the lines in an NMR coupling pattern are given by a binomial expansion or more conveniently by Pascal s triangle (Table 21.2). To derive Pascal s triangle, start at 1 and generate each lower row by adding together the two numbers above and to either side in the row above. 

TABLE 8.1 Pascal s Triangle (Coefficients of the Binomial Distribution)... 

The relative intensities obey the mathematical mnemonic known as Pascal s triangle. (You may remember this mnemonic from one of your math classes.) According to Pascal, each number at the bottom of a triangle in the rightmost column of Table 14.2 is the sum of the two numbers to its immediate left and right in the row above it. 

You should now be able to work out why the resonance for the OH proton is a triplet. Notice that the intensity of the peaks of the split resonances are not all the same - these approximately follow a set of numbers known as Pascal s triangle in which each number is the sum of the left and right numbers above it (Table 20.1 gives the first five lines of the triangle). 

IT = 0, 1,. . . , n in the binomial expansion. A device known as Pascal s triangle is the favorite method for doing this (Table LVIII). The rule for forming the values of the coefficients should be clear to the reader. 

However, even with the aid of Pascal s triangle, such computation can be excessive. Fortunately, tables of the binomial probability distribution have been prepared by the National Bureau of Standards (1950) which give sums such as we require for values of ti going from 0.01 to 0.50 (in steps of 0.01) and for values of n from 2 to 49 inclusive. 

Table 1 - Pascal s triangle each number is the sum of the two numbers above it.

Table 2 - Binomial coefficients in Pascal s triangle (revolved counter-clockv ise on 45 degrees).

Table 3 - Doubled binomial coefficients in a modification of Pascal s triangle.

The values of all numbers in Table 3 correspond to doubling the numbers in the conventional Pascal s triangle. From this modified table it is possible to obtain ... 

The method of finite differences, which is involved in the building of Pascal s triangle, can sometimes be used to conjecture a formula/(n). Table 4 shows the first and second differences obtained fi-om sequence (a). 

Note that the second differences are constant (i.e., 15). If and when we reach a row that contains a constant value, we can write an explicit expression for fin). In Pythagorean terms one would say that if the method of finite differences is applicable, each difference can be considered as a gnomon for a previous series. We can consider numbers in Table 4 as Pascal s triangle modification C . 

The relative intensities of the components in a multiplet signal are given by Pascal s triangle, which is based on the coefficients of the expansion of (a+b)". For proton spectra, the n+1 rule and Pascal s triangle lead to the midti-plicities and relative intensities for an observed resonance signal when there are n adjacent identical nuclei as shown in Table 5. 

Table 2 Line intensities for a series of nuclei with different values of I. For the case of / = the line intensities can be described by the expansion of x+ and hence follow Pascal s triangle...

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