Dyckpaths
A039599 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{2n}{n - k} (2k + 1) / (n + k + 1) } \)
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 3 1 5 9 1 7 20 28 1 9 35 75 90 1 11
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 1 2 1 5 3 14 9 1 42 28 5 132 90 20 1
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 3 2 9 10 3 28 40 21 4 90 150 105 36 5
\(T_{n + 3, n}\) ➤ TablDiag3 ➤ 5 28 75 154 273 440 663 950 1309 1748
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 6 45 140 3150 207900 21021 2522520
\(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 3 9 28 90 297 1001 3640 13260 48450
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 1 3 9 20 75 154 637 1260 5508 10659
\(T_{2 n + 1, n}\) ➤ CentralO ➤ 1 9 75 637 5508 48279 427570 3817125
\(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 0 -3 8 -5 -36 140 -176 -441 2600
\(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 2 12 104 1150 15492 247254 4577568
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 1 3 2 5 9 5 7 20 28 14 9 35 75 90 42 11
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 1 2 3 5 9 5 14 28 20 7 42 90 75 35 9
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 1 1 3 1 5 2 1 7 9 1 9 20 5 1 11
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 1 3 4 5 18 15 7 40 84 56 9 70 225 360
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 3 5 20 35 154 273 1260 2244 10659
\(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 5 35 273 2244 19019 164450 1442025
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 1 11 86 563 3314 18190 95052 478979
\(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 2 15 232 5725 197796 8859739