Domino Sums Take 2

A single domino is a rectangular tile divided into two square ends marked with spots. A domino set of order nn consists of all the domino tile combinations with spot counts from 00 to nn inclusive.

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

For example, a domino set of order 22 would consist of the domino tiles with numbers (0,0)(0, 0), (0,1)(0, 1), (0,2)(0, 2), (1,1)(1, 1), (1,2)(1, 2), (2,2)(2, 2), such that T(2)=(0+0)+(0+1)+(0+2)+(1+1)+(1+2)+(2+2)=12P(2)=(00)+(01)+(02)+(11)+(12)+(22)=7.\begin{aligned} 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{aligned} What is limnnT(n)P(n)?{\displaystyle \lim_{n\to\infty}} \frac{n T(n)}{P(n)}?

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