To solve this problem, let’s first break it down:

Step 1: Structure of Pascal's Triangle

The $n$-th row of Pascal's triangle is symmetrical and follows the binomial coefficient formula: $\text{Value at position } k \text{ in row } n = \binom{n}{k}$ The middle element(s) of each row are given by $\binom{n}{\lfloor n/2 \rfloor}$, where $\lfloor \cdot \rfloor$ is the floor function.

Since we are transforming the triangle by multiplying each element by 2 and then adding 3, the formula for the transformed value at position $k$ in row $n$ becomes: $\text{New value} = 2 \cdot \binom{n}{k} + 3$

Step 2: Middle Value in the 19th Row

The 19th row of the original Pascal’s triangle has 19 elements. The middle value will be at the 10th element (since the indices start from 0). This corresponds to: $\text{Middle value} = \binom{19}{9}$

Step 3: Compute $\binom{19}{9}$

Using the binomial coefficient formula: $\binom{n}{k} = \frac{n!}{k! \cdot (n-k)!}$ We get: $\binom{19}{9} = \frac{19!}{9! \cdot 10!} = 92378$

Step 4: Apply the Transformation

Now we multiply this value by 2 and add 3: $\text{Transformed middle value} = 2 \cdot 92378 + 3 = 184759$

Final Answer

The value at the center of your transformed 19th row is: $\boxed{184759}$

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Do you need any further clarifications or details?

Related Questions:

1. What is the sum of all elements in the 19th row of your transformed triangle?
2. How do you derive the symmetry property of Pascal’s triangle?
3. How would the transformation change if you added 5 instead of 3?
4. What is the ratio of the center element of the 19th row to the 18th row in the transformed triangle?
5. How many entries in the 19th row are divisible by 2 after transformation?

Tip:

When working with binomial coefficients, keep in mind their symmetry: $\binom{n}{k} = \binom{n}{n-k}$. This can simplify some calculations!