Tsao, Hung-ping 曹恆平 (2021). Evolutionary mathematics and science for Ultimate Generalization of Lah Numbers/(Binomial Coefficients): Sums/(Alternate Sums) of Orthogonal Products of Stirling Numbers. In: “Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)”, Wang, Lawrence K. 王抗曝 and Tsao, Hung-ping 曹恆平 (editors). Volume 3, Number 6, June 2021; 34 pages. Lenox Institute Press, Newtonville, NY, 12128-0405, USA. No. STEAM-VOL3-NUM6-JUN2021; ISBN 978-0-9890870-3-2. ……………ABSTRACT: We first introduce Stirling and Lah numbers via recursion and express Lah numbers and binomial coefficients as sums and alternate sums of orthogonal products of Stirling numbers of both kinds, respectively. After pointing out that Fibonacci numbers are nothing but upward diagonal sums of Pascal triangle, we generalize the triangular arrays in question from the natural sequence based to arithmetically progressive sequences based and call their upward diagonal sum Fibonacci values. After looking at more triangular arrays based on other sequences such as binomial coefficients and Fibonacci numbers, we eventually conclude that such construction of triangular arrays works with any underlying sequence base. ……………KEYWORDS: Binomial coefficient, Stirling number, Lah number, Sum, Alternate sum, Orthogonal product, Natural sequence, Arithmetically progressive sequence, Recursion, Fibonacci number, upward diagonal, q-Gaussian coefficient.