This paper studies properties of the integer sequence \(\overline{\overline{G}}_n=\prod _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) _{\mathbb {Z,N}}\) which is analogous to \(\overline{G}_n=\prod _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) \) , the product of the elements of the n-th row of Pascal’s triangle. Here \(\left( {\begin{array}{c}n\\ k\end{array}}\right) _{\mathbb {Z,N}}\) is an extended binomial coefficient, defined in the paper, constructed using an extended version of M. Bhargava’s theory of generalized factorials. In 1996 M. Bhargava introduced a generalization of the factorial function, \(n!_S=\prod _p\nu _n(S,p)\) in terms of their prime factorization, and defines associated binomial coefficients. The last two authors extended Bhargava’s invariants further to define such invariants attached to each integer \(b\ge 2\) . One obtains extended factorials and extended binomial coefficients, and the maximal extension defines extended factorials \(n!_{\mathbb {Z,N}}=\prod _{b\ge 2}b^{\alpha _n(\mathbb {Z},b)}\) including all \(b\ge 2\) , with associated extended binomial coefficients \(\left( {\begin{array}{c}n\\ k\end{array}}\right) _{\mathbb {Z,N}}\) , yielding \(\overline{\overline{G}}_n\) . We have \(\overline{\overline{G}}_n=\prod _{b=2}^nb^{\overline{\nu }(n,b)}\) and the partial factorizations \(\overline{\overline{G}}(n,x)=\prod _{b=2}^{\lfloor x\rfloor }b^{\overline{\nu }(n,b)}\) . This paper shows \(\log \overline{\overline{G}}(n,\alpha n)\) is well approximated by \(f_{\overline{\overline{G}}}(\alpha )n^2\log n+g_{\overline{\overline{G}}}(\alpha )n^2\) as \(n\rightarrow \infty \) for limit functions \(f_{\overline{\overline{G}}}(\alpha )\) and \(g_{\overline{\overline{G}}}(\alpha )\) defined for all \(0\le \alpha \le 1\) . The remainder term has a power saving in n. The main results are deduced from study of functions \(\overline{A}(n,x)\) and \(\overline{B}(n,x)\) that encode statistics of the base b radix expansions of the integer n (and smaller integers), where the base b ranges over all integers \(2\le b\le x\) . Unconditional estimates of \(\overline{A}(n,x)\) and \(\overline{B}(n,x)\) are derived.
