Stirlingcycb
A028338 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \sum_{i=k}^{n} (-2)^{n-i} \binom{i}{k} {n \brack i} } \)
\(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 4 1 23 9 1 176 86 16 1 1689 950 230
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 4 1 9 23 1 16 86 176 1 25 230 950
\(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -4 1 13 -9 1 -40 58 -16 1 121 -330
\(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -4 1 -9 13 1 -16 58 -40 1 -25 170
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 1 3 1 15 4 105 23 1 945 176 9 10395
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 1 2 3 7 8 15 38 47 48 105 281 367 383
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 4 2 23 18 3 176 172 48 4 1689 1900
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 12 1035 794640 2324992950
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 3 18 148 1520 18656 266112 4324608
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 3 14 92 784 8224 102528 1481472
\(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 1 4 19 129 1130 12171 155625 2303602
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 1 4 23 86 950 3480 57379 208054
\(T_{2 n + 1, n}\) ➤ CentralO ➤ 1 23 950 57379 4574934 453714470
\(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 3 14 92 784 8224 102528 1481472
\(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 8 68 680 7984 108416 1676800
\(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 2 12 112 1390 21316 387016 8089152
\(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 0 -4 28 -146 176 13096 -332856
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 1 4 3 9 23 15 16 86 176 105 25 230 950
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 1 3 4 15 23 9 105 176 86 16 945 1689
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 1 1 4 1 9 3 1 16 23 1 25 86 15 1
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 2 1 5 8 1 10 33 48 1 17 103 279 384
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 1 4 6 9 46 45 16 172 528 420 25 460
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 3 18 148 1520 18656 266112 4324608
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 4 9 86 230 3480 10045 208054 626934
\(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 9 230 10045 626934 51069018
\(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 3 18 148 1520 18656 266112 4324608
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 1 16 236 3624 60144 1089920 21578752
\(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 48 231 640 1365 2496 4123 6336 9225
\(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 2 21 640 39585 4133376 653309965