Notes

Chapter 10: Processes of Perception and Analysis

Section 11: Traditional Mathematics and Mathematical Formulas


Trinomial coefficients

The coefficient of xn in the expansion of (1 + x + x2)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]. This result follows directly from the generating function formula

(1 - 2 x z + x2)-mSum[GegenbauerC[n, m, z] xn, {n, 0, }]

From Stephen Wolfram: A New Kind of Science [citation]