Palindromes, Pascal Triangles, and Digitals Convolutions in Base b

We study even palindromes in base b through a new viewpoint based on discrete convolutions and Pascal-type transforms. Given an integer N, the associated palindrome T_b(N) can be interpreted as a digital autocorrelation of its base-b digits, while division by b+1 corresponds to a binomial transform perturbed by localized carry effects. We introduce the ideal quotient, obtained by enforcing symmetric carries, and show that its digits arise from a folded Pascal-triangle structure. Numerical experiments across several bases reveal unexpected regularity phenomena, including high palindromicity rates in the diagonal products U(N,N)=T_b(N)^2/(b+1) and universal “smooth” digit patterns. We formulate classification and probabilistic questions motivated by these findings.