And as the table below illustrates, the entries in Pascal's triangle are simply the binomial coefficients that appear when one expands out the powers of 1 + x . … These coefficients can also be obtained from the formulas in terms of Binomial and GegenbauerC given. … Binomial[m, n] is the ordinary binomial coefficient m!

Binomial distribution If black squares appear independently with probability p then the probability that m squares out of n are black is Binomial[n, m] p m (1 - p) n - m .

Trinomial coefficients The coefficient of x n in the expansion of (1 + x + x 2 ) t is Sum[Binomial[n + t - 1 - 3k, n - 3k] Binomial[t, k] (-1) k , {k, 0, t}] which can be evaluated as Binomial[2t, n] Hypergeometric2F1[-n, n - 2t, 1/2 - t, 1/4] or finally GegenbauerC[n, -t, -1/2] .

Pascal's triangle and rule 90 As shown on page 611 the pattern produced by rule 90 is exactly Pascal's triangle of binomial coefficients reduced modulo 2: black cells correspond to odd binomial coefficients. … The positions of the black cells are given by (and this establishes the connection with the picture on page 117 ) Fold[Flatten[{#1 - #2, #1 + #2}] &, {0}, 2^DigitPositions[t]] DigitPositions[n_] := Flatten[Position[Reverse[IntegerDigits[n, 2]], 1]] - 1 The actual pattern generated by rule 90 corresponds to the coefficients in PolynomialMod[Expand[(1/x + x) t ], 2] (see page 1091 ); the color of a particular cell is thus given by Mod[Binomial[t, (n + t)/2], 2] /; EvenQ[n + t] . Mod[Binomial[t, n], 2] yields a distorted pattern that is the one produced by rule 60 (see page 58 ).

The digit-based approach to finding binomial coefficients modulo k has been invented independently many times since the mid-1800s, notably by Edouard Lucas in 1877 and James Glaisher in 1899. The fact that the odd binomial coefficients form a nested geometrical pattern had apparently not been widely noticed before I emphasized it in 1982.

A related result is that IntegerExponent[Binomial[t, n], k] is given by the number of borrows in the base k subtraction of n from t . Mod[Binomial[t, n], k] is given for prime k by With[{d = Ceiling[Log[k, Max[t, n] + 1]]}, Mod[Apply[Times, Apply[Binomial, Transpose[ {IntegerDigits[t, k, d] , IntegerDigits[n, k, d] }], {1}]], k]] The patterns obtained for any k are nested.

Entropy of particles in a box The number of possible states of a region of m cells containing q particles is Binomial[m, q] .

The numbers in the top picture are binomial coefficients; those in the bottom picture are particular trinomial coefficients.

If k is irrational then equidistribution of Mod[Binomial[t, x], k] implies that all possible values eventually appear; the corresponding patterns seem fairly irregular, as shown below.

Continuous generalizations [of additive rules] Functions such as Binomial[t, n] and GegenbauerC[n, -t, -1/2] can immediately be evaluated for continuous t and n .