Can anyone solve the "PLANTERSPEANUTS Puzzle"?

Question

PLANTERSPEANUTS Puzzle: In the early 1950's the makers of Planters Peanuts sponsored a contest and offered a prize to anyone who could determine the number of ways to spell PLANTERSPEANUTS by moving either downward, to the left, to the right, or any combination of these directions in the fifteen-rowed pyramid shown in the figure below.

               P
              PLP
             PLALP
            PLANALP
           PLANTNALP
          PLANTETNALP
         PLANTERETNALP
        PLANTERSRETNALP
       PLANTERSPSRETNALP
      PLANTERSPEPSRETNALP
     PLANTERSPEAEPSRETNALP
    PLANTERSPEANAEPSRETNALP
   PLANTERSPEANUNAEPSRETNALP
  PLANTERSPEANUTUNAEPSRETNALP
 PLANTERSPEANUTSTUNAEPSRETNALP

Solve the PLANTERSPEANUTS puzzle by working parts (a) and (b).

(a) If $T_n: n=1,2,\ldots$ denotes the number of ways in which one can spell an $n$-lettered word in an $n$-rowed pyramid, show that $T_n$ satisfies:

$$ T_{n+1}=2T_n+1, T_1=1$$

(b) Solve the initial-value problem in part (a) and use this result to solve the PLANTERSPEANUTS puzzle.

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And here is my try:

I tried to form difference equation.

Assume that the no. of times an $n$-lettered word formed in and $n$-rowed pyramid is $T_n$

So:

$$\begin{align}T_1 = 1\\T_2 = 3\\T_3 = 7\\T_4 = 15\end{align}$$

and I tried to get an difference equation from this equations.

So I formed,

$$\begin{align}T_2 - T_1 = 2\\T_3 - T_2 = 2^2\\T_4 - T_3 = 2^3\\\vdots\\T_n - T_{n-1} = 2^{n-1}\end{align}$$

and I don't know how to proceed further.