Congruences, Stirling Transforms, Meixner Fractions, and Euler Products in Integer Sequences

We assemble exact results for integer sequences arising from diagonal exponential generating functions, negative-binomial moments and Stieltjes fractions, integral Stirling substitutions, signed logarithmic generating functions, and Euler products. Universal coefficient congruences and a combinatorial derivation of Hadamard closure give all-index statements, while a fixed-order logarithmic Stirling theorem establishes eventual periodicity modulo every positive integer. A uniform modulo-four calculation gives precise counterexamples to three proposed totient-period bounds without precluding eventual periodicity. For a companion Euler product we determine prime-power exponents, prove an all-index Jordan-totient approximation, and obtain logarithmic coefficient growth. A further inverse-Stirling and Witt-ghost construction supplies a quantitative prime-power transfer principle. Classical Meixner theory, Stirling identities, and earlier modular-periodicity results are explicitly distinguished from the deductions developed here. Complete, exact indexed examples and the reproducibility data accompanying this manuscript complement, but do not replace, the infinite arguments. Licensing note: The manuscript is licensed under CC BY 4.0. The OEIS b-file snapshots included in the supplementary archive retain their original CC BY-SA 4.0 license and attribution. The independently authored verification code is made available under CC BY 4.0.