Some properties of binomial coefficients and their application to growth modelling

Abstract

Some properties of diagonal binomial coefficients were studied in respect to frequency of their units' digits. An approach was formulated that led to the use of difference tables to predict if certain units' digits can appear in the values of binomial coefficients at quadratic terms of the binomial theorem. Frequency distributions of units' digits of binomial coefficients contain gaps (zero frequency) under most common numbering systems with supposed exclusion to systems with 2 n bases. In the work, an application of binomial coefficient arithmetic to model cell population dynamics was suggested. For a multicellular organism, the growth of number of cells was presented as a succession of binomial coefficients. It was inferred that the number of cells in a multicellular organism may obey power function laws.

Sequences of odd numbers, even numbers and integer squares: gaps in frequency distributions of unit's digits of minor totals

“…In this study, I considered a question as to whether all digits of a numbering system can appear as unit's digits in minor totals of i) the sequence of even integers, ii) sequence of odd integers, and in iii) Faulhaber polynomials with p = 2. Many integer sequences [2] have gappy frequency distributions of unit's digits in their minor totals. This means that some digits never appear as unit's digits in the minor totals.…”

“…Binomial coefficients have been extensively studied, the focus being often on divisibility of the coefficients by primes [1,3,4,5,6]. Recently Gavrikov [2] showed the binomial coefficients at quadratic terms (that is, minor totals of the natural sequence) to possess some other properties. The frequencies with which the unit's digits appear in the minor totals may be in part predicted-some digits never appear as unit's digits in the minor totals, i.e., they have zero frequencies.…”

“…In terms of modular arithmetic, the last propositions may be presented as S n ≡ A (mod 10), A ∈ {2, 4, 7, 9} and S n ≡ B (mod 8), B ∈ {0, 1, 2, 3, 4, 5, 6, 7}. Gavrikov [2] studied the presence or absence of zero frequencies (gaps) in frequency distribution of unit's digits of minor totals for numbering systems with bases 7 and 8, with an approach of "difference tables" being suggested. The approach theoretically allows one to predict the gaps in unit's digits for any numbering system.…”

Generating functions for series involving higher powers of inverse binomial coefficients and their applications

“…The binomial coefficient can also be used to construct mathematical model for the cell population dynamics and computation the growth of number of cells of multicellular organism (cf. 10 ).…”

Generating functions for series involving higher powers of inverse binomial coefficients and their applications

“…The binomial coefficient can also be used to construct mathematical model for the cell population dynamics and computation the growth of number of cells of multicellular organism (cf. [9]).…”