Domino Sums Take 2

A single domino is a rectangular tile divided into two square ends marked with spots. A domino set of order \(n\) consists of all the domino tile combinations with spot counts from \(0\) to \(n\) inclusive.

\(T(n)\) is the sum of the totals of the two spot counts on each tile in the domino set of order \(n\), and \(P(n)\) is the sum of the products of the two spot counts on each tile in the domino set of order \(n\).

For example, a domino set of order \(2\) would consist of the domino tiles with numbers \((0, 0)\), \((0, 1)\), \((0, 2)\), \((1, 1)\), \((1, 2)\), \((2, 2)\), such that \[\begin{align} T(2) &= (0 + 0) + (0 + 1) + (0 + 2) + (1 + 1) + (1 + 2) + (2 + 2) \\&= 12\\\\ P(2) &= (0 \cdot 0) + (0 \cdot 1) + (0 \cdot 2) + (1 \cdot 1) + (1 \cdot 2) + (2 \cdot 2) \\&= 7. \end{align}\] What is \({\displaystyle \lim_{n\to\infty}} \frac{n T(n)}{P(n)}?\)

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