Divisibility
A113704 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ [n=0 \text{ or } k \text{ divides } n] } \)
A000005 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2
A000007 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
A000012 \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 -1 0 1 0 -1 0 1 0 -1 0 0 1 0 0
---------- \(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -1 1 0 -1 1 0 -1 0 1 0 0 0 -1 1 0 0
---------- \((T_{n + 1, n - k + 1})^{-1}\) ➤ Tinvrev11 ➤ 1 -1 1 -1 0 1 0 0 -1 1 -1 0 0 0 1 0 0 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
---------- \(T_{n , n}\) ➤ TablDiag0 ➤ 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
---------- \(T_{2 n, n}\) ➤ CentralE ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ RevPolyRow1 ➤ 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 ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A000035 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
---------- \(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
A000203 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 3 4 7 6 12 8 15 13 18 12 28 14 24
A001157 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 5 10 21 26 50 50 85 91 130 122 210
A001227 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 1 2 1 2 2 2 1 3 2 2 2 2 2 4 1 2 3 2
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 1 0 1 0 2 0 1 0 2 0 2 0 2 0 1 0 3 0
---------- \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 2 1 2 2 2 1 3 2 2 2 2 2 4 1 2 3 2
A002378 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 2 6 12 20 30 42 56 72 90 110 132 156
A002522 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 2 5 10 17 26 37 50 65 82 101 122 145
A007503 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 5 6 10 8 16 10 19 16 22 14 34 16 28
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 5 6 10 8 16 10 19 16 22 14 34 16 28
A032741 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 1 2 1 3 1 3 2 3 1 5 1 3 3 4 1 5 1
A034262 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 2 10 30 68 130 222 350 520 738 1010
A051731 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 1 1 1 0 1 1 1 0 1 1 0 0 0 1 1 1 1 0 0
A055895 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 6 10 22 34 78 130 278 522 1062 2050
A056045 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 3 4 11 6 42 8 107 94 308 12 1718 14
A066108 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 6 30 276 3130 46914 823550 16781384
A074854 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 3 5 13 17 57 65 209 321 801 1025
A079978 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0
A081307 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 3 4 8 6 16 8 21 17 26 12 50 14 36
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 3 4 8 6 16 8 21 17 26 12 50 14 36
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 3 4 8 6 16 8 21 17 26 12 50 14 36
A094471 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 1 2 5 4 12 6 17 14 22 10 44 12 32
A112329 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 0 -2 1 -2 0 -2 2 -3 0 -2 2 -2 0 -4
A113704 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 1 0 0 0
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 -1 1 0 -1 0 -1 0 -1 1 0 1 0 -1
A113998 \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 1 1 0 1 1 0 1 1 1 0 0 0 1 1 0 0 1 1
A127093 \(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 1 2 1 0 3 1 2 0 4 1 0 0 0 5 1 2 3 0 0
A128174 \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
A174852 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -1 1 -1 0 1 0 -1 0 1 -1 0 0 0 1 1 -1
A175992 \(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 0 0 1 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0
A183063 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 1 0 2 0 2 0 3 0 2 0 4 0 2 0 4 0 3 0
A320111 \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 1 2 2 2 2 2 3 3 2 2 4 2 2 4 4 2 3 2
A357051 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 4 10 37 82 352 730 2998 7291 26488
A363913 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 12 30 93 246 768 2190 6654 19713
A363914 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 0 -1 1 0 -1 0 1 0 0 -1 0 1 0 -1 0
A367326 \(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 0 1 4 13 16 50 36 101 100 170 100 402