Binomialbell
A056857 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k} \text{Bell}(n-k) } \)
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 2 1 3 6 1 4 12 20 1 5 20 50 75 1 6
\(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -2 1 0 -3 1 4 0 -4 1 5 10 0 -5 1 -12
\(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -2 1 -3 0 1 -4 0 4 1 -5 0 10 5 1 -6
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 1 2 1 5 2 15 6 1 52 20 3 203 75 12 1
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 1 2 2 4 5 5 11 14 15 15 35 47 51 52
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 2 2 6 6 3 20 24 12 4 75 100 60 20 5
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 2 30 60 3900 4750200 4860246300
\(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 2 1 1 1 1 1 2 3 1 1 1 1 1 1 1 1 1 1
\(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 2 6 20 75 312 1421 7016 37260
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 3 11 45 200 958 4921 26981 157203
\(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 1 3 7 22 75 291 1243 5807 29360
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 1 2 6 12 50 100 525 1050 6552 13104
\(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 2 7 33 184 1153 7937 59188 473305
\(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 0 -1 5 -8 -1 121 -778 3417 -10211
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 1 2 2 3 6 5 4 12 20 15 5 20 50 75 52 6
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 1 2 2 5 6 3 15 20 12 4 52 75 50 20 5
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 1 1 2 1 3 2 1 4 6 1 5 12 5 1 6 20
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 2 1 3 5 1 4 10 15 1 5 17 37 52 1 6
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 1 2 4 3 12 15 4 24 60 60 5 40 150 300
\(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 3 7 28 96 459 1989 10931 56275
\(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 1 2 8 24 107 418 2151 10216 59700
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 3 11 45 200 958 4921 26981 157203
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 2 3 12 20 100 175 1050 1890 13104
\(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 3 20 175 1890 24024 348348 5643495
\(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 3 11 45 200 958 4921 26981 157203
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 1 10 72 472 3035 19734 131579 905720
\(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 15 71 199 429 791 1315 2031 2969 4159