Catalanpaths
A053121 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ is(k = 0)\ ? \ 0 : \frac{k+1}{n+1} \binom{n+1}{(n-k)/2} } \)
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 0 1 0 2 1 0 3 0 1 0 4 0 5 1 0 5 0 9
\(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 0 1 0 -2 1 0 -3 0 1 0 -4 0 3 1 0 -5
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 0 1 1 0 0 2 2 1 0 0 0 5 5 3 1 0 0 0 0
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 0 1 1 1 2 0 2 2 3 2 2 5 5 6 0 5 5 9 9
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 0 2 2 0 3 0 6 0 4 5 0 12 0 5 0 18 0
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 1 2 6 20 45 42 140 3024 3150 660
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 0 2 3 0 0 14 20 0 0 110 154 0 0 910
\(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 4 11 28 66 152 339 748 1622 3496
\(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 5 33 306 3650 53465 929285 18695950
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 0 0 1 0 2 0 0 3 0 2 0 4 0 5 0 0 5 0 9 0
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 0 1 0 0 2 0 2 0 3 0 0 5 0 4 0 5 0 9 0 5
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 0 1 0 1 0 1 1 0 2 1 0 3 0 1 0 4 0
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 0 0 2 0 4 0 0 6 0 8 0 8 0 20 0 0 10 0
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 0 0 3 4 0 0 20 27 0 0 154 208 0 0
\(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 2 4 14 28 76 152 374 748 1748 3496
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 0 4 8 44 96 344 752 2252 4880 13256
\(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 5 19 561 3226 245125 1680995