iterated forward difference operator applied to primes, OEIS A007442

I apply the iterated forward difference operator to the sequence of primes; from

$$ (p_n) = (2, 3, 5, 7, 11 \ldots) $$

I get

$$ (d^1_n) = (1, 2, 2, 4 \ldots) $$ $$ (d^2_n) = (1, 0, 2 \ldots) $$ $$ (d^3_n) = (-1, 2 \ldots) $$ $$ \ldots $$

For each sequence $(x_n)$, one can reconstruct the sequence from the first entries of the iterated forward differences $(d^0_1 = x_1, d^1_1 = x_2 - x_1, d^2_1 = d^1_2 - d^1_1, \ldots)$:

So from the sequence

$$ (d^n_1) = (2, 1, 1, -1, 3, -9 \ldots) $$

one can reconstruct the primes $(p_n)$.

OEIS has a corresponding entry A007442 "Inverse binomial transform of primes".

As one can see here, the sequence grows very fast; numerically (using all primes up to 100000) I found an asymptotic behavior $ \sim 2^n $ which matches the largest binomial coefficient used for reconstruction the primes, i.e. $ C(n, \lfloor n / 2 \rfloor) $, neglecting sub-exponential terms.

These huge numbers cancel and produce small primes. So $p_{31} = 127$ follows from $n, k, C(n, k), (d^n_1)$:

30 0 1 2
30 1 30 1
30 2 435 1
30 3 4060 -1
30 4 27405 3
30 5 142506 -9
30 6 593775 23
30 7 2035800 -53
30 8 5852925 115
30 9 14307150 -237
30 10 30045015 457
30 11 54627300 -801
30 12 86493225 1213
30 13 119759850 -1389
30 14 145422675 445
30 15 155117520 3667
30 16 145422675 -15081
30 17 119759850 41335
30 18 86493225 -95059
30 19 54627300 195769
30 20 30045015 -370803
30 21 14307150 652463
30 22 5852925 -1063359
30 23 2035800 1570205
30 24 593775 -1961755
30 25 142506 1560269
30 26 27405 1401991
30 27 4060 -11023119
30 28 435 36000427
30 29 30 -93408425
30 30 1 214275735

As one can see, the sign changes in $(d^n_1)$ occur mostly between neighbored elements.

I eliminate the fast oscillation and the asymptotic behavior by defining

$$ D^n_1 = (-1)^n \cdot 2^{-n} \cdot d^n_1 $$

The result looks like this

Question: Is this all just a curiosity? Or are there any relevant results and open questions in number theory related to this sequence?