The paper introduces an infinite integer sequence progression that produces a triangular array for which the $n$th row sums to $2^n$ for every $n\geq0$. Every row of the triangular array can be realized as the degree sequence of a multigraph without loops. Although the first four rows coincide with those from Pascal's triangle, the proposed array progression becomes asymmetric and diverges from binomial coefficients thereafter. The triangular array hosts infinitely many unimodal \emph{near}-log-concave sequences with no zeros, and its log-concavity deviation converges to $\log(4/3)$ under logarithmic scaling. The progression provides a probability measure at every generation, defines what we call near-log-concave random variables, produces state-transitions of an infinite Markov chain, and hosts a totally-positive Toeplitz matrix of order two. We study the Shannon entropy dynamics of the construct, and present an application in physics to model systems exhibiting a set of state-transition characteristics that govern jumps of electrons across different energy states. We provide a transformation of the triangular array over the Tychonoff cube to produce an infinite family of right-stochastic matrices, all satisfying the given set of state-transition characteristics.
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