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4 two missprints corrected

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 $$\sum_{k=0}^{p-1}{2k\choose k}\frac{k^j}{k+1}\equiv \alpha_j+p\beta_j\pmod{p^2}$$ for every prime number $p\equiv 1\pmod 6$ and $$\sum_{k=0}^{p-1}{2k\choose k}\frac{k^j}{k+1}\equiv -(-1)^j-\alpha_j+p\beta_j\pmod{p^2}$$ for every prime number $p\equiv 5\pmod 6$.

(More precisely, the sequences $3^n\alpha_n$ and $3^n\beta_n$ 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 $n\geq 2$) seems to converge very quickly towards $-\frac{4\sqrt{3}\pi}{9}=-2.4183991523\dots$ (the error is smaller than $10^{-79}$ 10^{-78}$for$n=120$). The sequence$\frac{\alpha_{n+1}}{\alpha_n}-\frac{\alpha_n}{\alpha_{n-1}}$converges perhaps (fairly slowly) to towards something like$-.72\dots$. 3 Update with resuming my current knowledge added I am curious if somebody can be helpful concerning the following experimental observation: There exist two rational sequences$\alpha_0,\alpha_1,\dots$and$\beta_0,\beta_1,\dots$, both with values in$\mathbb Z[1/3]$such that $$\sum_{k=0}^{p-1}{2k\choose k}\frac{k^j}{k+1}\equiv \alpha_j+p\beta_j\pmod{p^2}$$ for every prime number$p\equiv 1\pmod 6$and $$\sum_{k=0}^{p-1}{2k\choose k}\frac{k^j}{k+1}\equiv -(-1)^j-\alpha_j+p\beta_j\pmod{p^2}$$ for every prime number$p\equiv 5\pmod 6$. (More precisely, the sequences$3^n\alpha_n$and$3^n\beta_n$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$n\geq 2$) seems to converge very quickly towards$-\frac{4\sqrt{3}\pi}{9}=-2.4183991523\dots$(the error is smaller than$10^{-79}$for$n=120$). The sequence$\frac{\alpha_{n+1}}{\alpha_n}-\frac{\alpha_n}{\alpha_{n-1}}$converges perhaps (fairly slowly) to something like$-.72\dots$. 2 Error in formula corrected and sentence enclosed in parentheses added 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 $$\sum_{k=0}^{p-1}{2k\choose k}\frac{k^j}{k+1}\equiv \alpha_j+p\beta_j\pmod{p^2}$$ for every prime number$p\equiv 1\pmod 6$and $$\sum_{k=0}^{p-1}{2k\choose k}\frac{k^j}{k+1}\equiv (-1)^j-\alpha_j+p\beta_j\pmod{p^2}$$ -(-1)^j-\alpha_j+p\beta_j\pmod{p^2}$$for every prime number p\equiv 5\pmod 6. (More precisely, the sequences 3^n\alpha_n and 3^n\beta_n are seemingly integral.) The sequence \alpha_0,\alpha_1,\dots starts as$$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.

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