Stirling's Approximation for Central Extended Binomial Coefficients

We derive asymptotic formulae for central polynomial coefficients, a generalization of binomial coefficients, using the distribution of the sum of independent uniform random variables and the CLT. 1 Stirling’s formula and binomial coefficients For a nonnegative integer m, Stirling’s formula m! ∼ √ 2πm (m e )m , where e is Euler’s number, yields an approximation of the central binomial coefficient ( m m/2 ) (assuming that m is even) using ( m k ) = m! k!(m−k)! as ( m m/2 ) ∼ 2 m+1 √ 2πm , where we write am ∼ bm as a short-hand for limm→∞ am bm = 1. In the current note, we derive an asymptotic formula for the central multinomial triangle or, polynomial, coefficient (cf. [1], [2]), where multinomial triangles are a generalization of binomial triangles, where entries in row k are defined as coefficients of the polynomial (1 + x+ x + . . .+ x) for l ≥ 0. Our derivation is not based upon asymptotics of factorials but upon the limiting distribution of the sum of discrete uniform random variables. Polynomial coefficients are important, for example, because they denote the number of integer compositions of the nonnegative integer n with parts in the set A := {a, a + 1, . . . , b}, where a, b, 0 ≤ a ≤ b, are nonnegative integers (cf. [3]), i.e. the number of possibilities to write n as a sum of integers p1, . . . , pk, with pi ∈ A, for i = 1, . . . , k. 2 Multinomial triangles In generalization to binomial triangles, (l + 1)-nomial triangles, l ≥ 0, are defined in the following way. Starting with a 1 in row zero, construct an entry in row k, k ≥ 1, by adding the overlying (l + 1) entries in row (k − 1) (some of these entries are taken as zero if not defined); thereby, row k has (kl + 1) entries. For example, the monomial (l = 0), binomial (l = 1), trinomial (l = 2) and quadrinomial triangles (l = 3) start as follows, 1 1 1 1 1 1 1 1 2 1 1 3 3 1 1 1 1 1 1 2 3 2 1 1 3 6 7 6 3 1 1 1 1 1 1 1 2 3 4 3 2 1 1 3 6 10 12 12 10 6 3 1 1Throughout, we assume that m or l are even. If this is not the case, replace respective quantities, e.g. ml 2 , with their floor, b 2 c. 1 ar X iv :1 20 3. 21 22 v1 [ m at h. PR ] 9 M ar 2 01 2 In the (l+1)-nomial triangle, entry n, 0 ≤ n ≤ kl, in row k, which we denote by ( k n ) l+1 , has the following interpretation. It is the coefficient of x in the expansion of (1 + x+ x + . . .+ x) = kl ∑