nt.number theory - Modular congruences related to sums of Catalan numbers

I am curious if somebody can be helpful concerning the following
experimental observation:

There exist two rational sequences $alpha_0,alpha_1,dots$ and
$beta_0,beta_1,dots$, both with values in $mathbb Z[1/3]$
such that
$$sum_{k=0}^{p-1}{2kchoose k}frac{k^j}{k+1}equiv alpha_j+pbeta_jpmod{p^2}$$
for every prime number $pequiv 1pmod 6$ and
$$sum_{k=0}^{p-1}{2kchoose k}frac{k^j}{k+1}equiv -(-1)^j-alpha_j+pbeta_jpmod{p^2}$$
for every prime number $pequiv 5pmod 6$.

(More precisely, the sequences $3^nalpha_n$ and $3^nbeta_n$ are seemingly integral.)

The sequence $alpha_0,alpha_1,dots$ starts as
$$1, 0, -2/3, 4/3, -22/9, 140/27, -14, 1316/27, -17078/81, 87860/81, -1562042/243, 31323292/729, dots$$
and the first terms $beta_0,beta_1,dots$ are
$$0,0,2/3,-2,14/3,-34/3,98/3,-350/3,1526/3,-2622,46634/3,-311734/3,2316158/3, -18920018/3,dots$$

Let me end by remarking that one has as a special case a similar result when replacing Catalan numbers by central binomial coefficients.

Update: The existence of the sequence $alpha_n$ is explained by the Zhi-Wei Sun paper,
see the answer by dke below.

Experimentally, the quotient sequence $frac{beta_n}{alpha_n}$ (defined for $ngeq 2$)
seems to converge very quickly towards $-frac{4sqrt{3}pi}{9}=-2.4183991523dots$ (the error is
smaller than $10^{-78}$ for $n=120$).

The sequence $frac{alpha_{n+1}}{alpha_n}-frac{alpha_n}{alpha_{n-1}}$
converges perhaps (fairly slowly) towards something like $-.72dots$.

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