Does Tartaglia's Triangle Hide an Infinite Secret?

Imagine Tartaglia's Triangle (or Pascal's): a pyramid of numbers where each is the sum of the two above. Mathematicians love it because it pops up everywhere, from probability to algebra. But there is much more beneath the surface.

Take the "Pascal matrix", an infinite triangular version of this scheme. If you compute the determinants of all possible k×k squares inside, you get new numerical arrays called **Pascal Determinantal Arrays (PD_k)**. For k=1 you find the classic triangle; for k=2, 3... you discover mysterious new structures.

Here comes the magic. An elegant equation links these arrays: the "log-concavity" operator (measuring how much a sequence curves downward, like a² ≥ b×c) applied to PD_k gives exactly the element-wise product of PD_{k-1} and PD_{k+1}. Formula: **LC(PD_k) = PD_{k-1} ⊙ PD_{k+1]**.

The proof uses a classic trick: the Desnanot-Jacobi identity (Dodgson condensation), a method to compute determinants by breaking them into smaller pieces, applied at two adjacent positions.

Why does it matter? Because this factorization, combined with a new general inequality (the LC operator is "submultiplicative" for Hadamard products), proves that **every row of every PD_k is infinitely log-concave**. You can apply the LC operator forever and the numbers stay positive. It extends a famous theorem (McNamara-Sagan) from the simple triangle to the whole determinantal hierarchy.

Bonus result: the sequence of determinants for increasing k is log-convex. Math is sometimes more beautiful than a fractal.

The paper (arXiv:2512.06414v1) studies Pascal Determinantal Arrays $\PD_k$, defined as the $k \times k$ minors of the lower-triangular Pascal matrix $P = (\binom{a}{b})_{a,b \ge 0}$. The central result is an exact factorization of the row-wise log-concavity operator:
\[
\LC(\PD_k) = \PD_{k-1} \Had \PD_{k+1},
\]
where $\LC(a)_j = a_j^2 - a_{j-1}a_{j+1}$ and $\Had$ denotes the Hadamard (entrywise) product.

The proof relies on an elementary manipulation of the Desnanot-Jacobi identity (Dodgson condensation) applied at two adjacent positions in the matrix. This classical identity relates the determinant of a matrix to those of its submatrices.

A general inequality is then proven: for log-concave arrays $A$ and $X$, the $\LC$ operator is submultiplicative with respect to the Hadamard product:
\[
\LC(A \Had X) \ge \LC(A) \Had \LC(X).
\]
Combining the factorization with this inequality yields a uniform algebraic proof that **every row of every array $\PD_k$ ($k \ge 1$) is infinitely log-concave**. This extends the celebrated theorem of McNamara and Sagan (which held for $\PD_1$, Pascal's triangle) to the entire determinantal hierarchy.

Applications include the log-convexity of $\{\PD_k(i,j)\}_{k \ge 0}$ in the determinantal order $k$ and a family of determinantal Hadamard inequalities.