Concatenating Pascal's Triangle

Calculus Level 5

Consider the following sequence (A003590): \[1, 11, 121, 1331, 14641, 15101051, 1615201561, \ldots\]

where the \(n^\text{th}\) term is the concatenation of all integers in the \(n^\text{th}\) row of the Pascal's Triangle in that order, beginning with \(n=0\).

Let \(L_n\) be the number of digits in the \(n^\text{th}\) term of this sequence.

Compute \(\displaystyle \lim_{n\to\infty}\exp\left(\frac{n^2}{L_n}\right) \).

Notation: \(\exp(x) \) denotes the exponential function, \(\exp(x) = e^x \).

Image Credit: N. J. A. Sloane

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