Worpitzky
A028246 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \sum_{j=0}^{n} \text{Eulerian}(n, j) \binom{n-j}{n-k} } \)
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 2 3 6 12 7 24 60 50 15 120 360 390
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 1 1 1 1 3 1 7 2 1 15 12 1 31 50 6 1
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 1 2 1 4 6 1 8 20 26 1 16 66 126 150 1
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 3 4 7 24 18 15 100 180 96 31 360 1170
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 6 84 600 145080 2167200
\(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 3 12 60 390 3360 31920 332640
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 3 11 55 359 2891 27635 305439 3829439
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 1 3 7 50 180 2100 10206 166824
\(T_{2 n + 1, n}\) ➤ CentralO ➤ 1 7 180 10206 1020600 158838240
\(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 11 109 1139 13021 164051 2273629
\(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 1 26 111 292 605 1086 1771 2696 3897
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 1 3 1 12 7 1 60 50 15 1 360 390 180 31
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 1 1 3 1 7 12 1 15 50 60 1 31 180 390
\(T^{-1}_{n + 1, n-k}\) ➤ RevTinv11 ➤ 1 -3 1 9 -7 1 -45 55 -15 1 585 -835 285
\(T^{-1}_{n + 1, n-k}\) ➤ RevTrevinv11 ➤ 1 1 -3 1 -7 9 1 -15 55 -45 1 -31 285
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 2 1 6 3 24 12 1 120 60 7 720 360 50
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 1 3 2 12 14 3 60 100 45 4 360 780 540
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 3 11 55 359 2891 27635 305439 3829439
\(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 3 9 37 187 1131 7965 64021 578371
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 3 12 50 390 2100 25200 166824
\(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 12 390 25200 2739240 451725120
\(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 1 5 29 209 1809 18269 210853 2737769
\(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 3 11 55 359 2891 27635 305439 3829439
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 1 7 49 411 4061 46103 590857 8434723
\(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 6 26 66 132 230 366 546 776 1062 1410
\(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 2 12 132 2280 56670 1907136 83094536