One
A000012 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ 1 } \)
A000007 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
A000012 \(T_{n, k}\) ➤ Triangle ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n, n - k}\) ➤ Trev ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 1
---------- \(T_{n , 0}\) ➤ TablCol0 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n , n}\) ➤ TablDiag0 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{2 n, n}\) ➤ CentralE ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{2 n + 1, n}\) ➤ CentralO ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 1
---------- \(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A000027 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
---------- \(\sum_{k=0}^{1} T_{1, k}\ n^k\) ➤ PolyRow1 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
---------- \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ RevPolyRow1 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A000035 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
A000079 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 2 4 8 16 32 64 128 256 512 1024 2048
A000217 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 3 6 10 15 21 28 36 45 55 66 78 91 105
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 3 6 10 15 21 28 36 45 55 66 78 91 105
---------- \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 3 6 10 15 21 28 36 45 55 66 78 91
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 3 6 10 15 21 28 36 45 55 66 78 91 105
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 3 6 10 15 21 28 36 45 55 66 78 91 105
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 1 3 6 10 15 21 28 36 45 55 66 78 91
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 3 6 10 15 21 28 36 45 55 66 78 91 105
A000225 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 3 7 15 31 63 127 255 511 1023 2047
---------- \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 3 7 15 31 63 127 255 511 1023 2047
A000330 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 5 14 30 55 91 140 204 285 385 506
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 1 5 14 30 55 91 140 204 285 385 506
A001045 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -1 3 -5 11 -21 43 -85 171 -341 683
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -1 3 -5 11 -21 43 -85 171 -341 683
A002061 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 1 3 7 13 21 31 43 57 73 91 111 133 157
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 3 7 13 21 31 43 57 73 91 111 133 157
A002260 \(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5
---------- \(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5
---------- \(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5
---------- \(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5
A003462 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 4 13 40 121 364 1093 3280 9841 29524
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 4 13 40 121 364 1093 3280 9841 29524
A004526 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10
---------- \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9
---------- \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9
---------- \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10
A031973 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 2 7 40 341 3906 55987 960800 19173961
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 2 7 40 341 3906 55987 960800 19173961
A053698 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 1 4 15 40 85 156 259 400 585 820 1111
---------- \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 4 15 40 85 156 259 400 585 820 1111
A097806 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 -1 1 0 -1 1 0 0 -1 1 0 0 0 -1 1 0 0 0
---------- \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 -1 1 -1 0 1 -1 0 0 1 -1 0 0 0 1 -1
---------- \((T_{n, n - k})^{-1}\) ➤ Tinvrev ➤ 1 -1 1 0 -1 1 0 0 -1 1 0 0 0 -1 1 0 0 0
---------- \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -1 1 0 -1 1 0 0 -1 1 0 0 0 -1 1 0 0 0
---------- \(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -1 1 -1 0 1 -1 0 0 1 -1 0 0 0 1 -1
---------- \((T_{n + 1, n - k + 1})^{-1}\) ➤ Tinvrev11 ➤ 1 -1 1 0 -1 1 0 0 -1 1 0 0 0 -1 1 0 0 0
---------- \(T^{-1}_{n + 1, n-k}\) ➤ RevTinv11 ➤ 1 -1 1 0 -1 1 0 0 -1 1 0 0 0 -1 1 0 0 0
---------- \(T^{-1}_{n + 1, n-k}\) ➤ RevTrevinv11 ➤ 1 1 -1 1 -1 0 1 -1 0 0 1 -1 0 0 0 1 -1
---------- \((T_{n + 1, n-k + 1})^{-1}\) ➤ RevTinvrev11 ➤ 1 -1 1 0 -1 1 0 0 -1 1 0 0 0 -1 1 0 0 0