Eulerian
A173018 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \sum_{j=0}^{k} (-1)^{j} \binom{n+1}{j} (k+1-j)^{n} } \)
\(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 0 -1 1 0 3 -4 1 0 -23 33 -11 1 0
\(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 -1 0 1 -4 3 0 1 -11 33 -23 0 1
\(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -1 1 -4 3 1 -11 33 -23 1 -26 220
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 0 0 1 0 1 0 1 1 0 1 4 0 1 11 1 0 1 26
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 0 1 0 1 2 0 1 5 6 0 1 12 23 24 0 1 27
\(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 1 4 11 2 1 1 1 2 1 1 1 1 1 1 1 2 1
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 1 1 11 26 302 1191 15619 88234
\(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 1 26 1191 88234 9738114 1505621508
\(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 5 26 160 1140 9240 84000 846720
\(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 6 26 66 132 230 366 546 776 1062 1410
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 0 1 0 4 1 0 11 11 1 0 26 66 26 1 0 57
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 0 0 1 0 1 4 0 1 11 11 0 1 26 66 26 0 1
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 1 1 2 2 1 5 6 6 1 12 23 24 24 1 27
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 0 1 0 4 2 0 11 22 3 0 26 132 78 4 0 57