Ctree
A106465 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ is(n \text{ odd})\ ? \ 1 : (k + 1) \text{ mod } 2 } \)
A000012 \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 -1 1 0 -1 1 -1 -1 1 1 0 -1 0 0 1 -1
---------- \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 0 1 -1 -1 1 0 -1 0 1 0 1 -1 -1 1 0 0
---------- \(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 0 1 -1 -1 1 0 -1 0 1 -1 -1 1 0 1 0
---------- \(T_{n , 0}\) ➤ TablCol0 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n , n}\) ➤ TablDiag0 ➤ 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
---------- \(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ 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
---------- \(T_{2 n + 1, n}\) ➤ CentralO ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T^{-1}_{n + 1, n-k}\) ➤ RevTinv11 ➤ 1 0 1 -1 -1 1 0 -1 0 1 0 1 -1 -1 1 0 0
---------- \(T^{-1}_{n + 1, n-k}\) ➤ RevTrevinv11 ➤ 1 1 0 1 -1 -1 1 0 -1 0 1 -1 -1 1 0 1 0
---------- \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A000027 \(\sum_{k=0}^{1} T_{1, k}\ n^k\) ➤ 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 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
---------- \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
---------- \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
---------- \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
---------- \(T_{2 n, n}\) ➤ CentralE ➤ 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
A002522 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 1 2 5 10 17 26 37 50 65 82 101 122 145
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 2 5 10 17 26 37 50 65 82 101 122 145
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
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10
A029578 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9
A052992 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -1 5 -5 21 -21 85 -85 341 -341 1365
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -1 5 -5 21 -21 85 -85 341 -341 1365
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
A056136 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 2 6 6 15 12 28 20 45 30 66 42 91 56
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 1 2 6 6 15 12 28 20 45 30 66 42 91 56
A103424 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 0 2 0 8 0 32 0 128 0 512 0 2048 0
A106465 \(T_{n, k}\) ➤ Triangle ➤ 1 1 1 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1
---------- \(T_{n, n - k}\) ➤ Trev ➤ 1 1 1 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 1 -1 1 0 1 1 -1 1 -1 1 0 1 0 1 1 -1 1
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -1 1 0 1 1 -1 1 -1 1 0 1 0 1 1 -1 1
A106466 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 1 2 1 3 2 4 2 5 3 6 3 7 4 8 4 9 5 10
---------- \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 2 1 3 2 4 2 5 3 6 3 7 4 8 4 9 5 10
A106468 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 -1 1 -1 0 1 1 -1 -1 1 0 0 -1 0 1 0 0
---------- \((T_{n, n - k})^{-1}\) ➤ Tinvrev ➤ 1 -1 1 -1 0 1 1 -1 -1 1 0 0 -1 0 1 0 0
A142150 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0
---------- \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 10
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0
A158302 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 2 2 8 8 32 32 128 128 512 512 2048
A166486 \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1
---------- \(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1
A336676 \(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 0
---------- \(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 0
A359366 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 3 4 10 9 21 16 36 25 55 36 78 49 105
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 3 4 10 9 21 16 36 25 55 36 78 49 105
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 3 4 10 9 21 16 36 25 55 36 78 49 105
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 3 4 10 9 21 16 36 25 55 36 78 49 105
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 3 4 10 9 21 16 36 25 55 36 78 49 105
A362448 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 0 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 0 1 0
---------- \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 0 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 0 1
---------- \(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 1 0 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 0 1 0
---------- \(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 1 1 0 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 0 1