Ordinals
A002262 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ k } \)
A000007 \(T_{n , 0}\) ➤ TablCol0 ➤ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
A000012 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 2 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 \(T_{n , n}\) ➤ TablDiag0 ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(T_{2 n, n}\) ➤ CentralE ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(\sum_{k=0}^{1} T_{1, k}\ n^k\) ➤ PolyRow1 ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
A000217 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 0 1 3 6 10 15 21 28 36 45 55 66 78 91
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 0 1 3 6 10 15 21 28 36 45 55 66 78 91
A000292 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 0 1 4 10 20 35 56 84 120 165 220 286
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 0 1 4 10 20 35 56 84 120 165 220 286
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 1 4 10 20 35 56 84 120 165 220 286
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 0 1 4 10 20 35 56 84 120 165 220 286
A000295 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 0 1 4 11 26 57 120 247 502 1013 2036
A000330 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 5 14 30 55 91 140 204 285 385 506
A000340 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 0 1 5 18 58 179 543 1636 4916 14757
A000537 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 9 36 100 225 441 784 1296 2025 3025
A001787 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 0 1 4 12 32 80 192 448 1024 2304 5120
A002260 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5
A002262 \(T_{n, k}\) ➤ Triangle ➤ 0 0 1 0 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 4
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 0 0 -1 0 -1 2 0 -1 2 -3 0 -1 2 -3 4 0
---------- \(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 0 0 1 0 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 4
A002415 \(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 0 1 6 20 50 105 196 336 540 825 1210
A002620 \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 0 1 2 4 6 9 12 16 20 25 30 36 42 49 56
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 1 2 4 6 9 12 16 20 25 30 36 42 49
---------- \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 0 1 2 4 6 9 12 16 20 25 30 36 42 49 56
A003418 \(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 2 6 12 60 60 420 840 2520 2520
A004526 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 0 -1 1 -2 2 -3 3 -4 4 -5 5 -6 6 -7 7 -8
---------- \(T_{n, n / 2}\) ➤ ColMiddle ➤ 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9
---------- \(T_{n, n / 2}\) ➤ RevColMiddle ➤ 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9
A004736 \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 2 1 3 2 1 4 3 2 1 5 4 3 2 1 6 5 4 3 2
A007290 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 0 2 8 20 40 70 112 168 240 330 440 572
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 0 2 8 20 40 70 112 168 240 330 440 572
A008794 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 1 4 4 9 9 16 16 25 25 36 36 49 49
A008805 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 0 0 1 1 3 3 6 6 10 10 15 15 21 21 28 28
A010701 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
A014105 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 3 10 21 36 55 78 105 136 171 210 253
A025581 \(T_{n, n - k}\) ➤ Trev ➤ 0 1 0 2 1 0 3 2 1 0 4 3 2 1 0 5 4 3 2 1
---------- \(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 0 1 0 2 1 0 3 2 1 0 4 3 2 1 0 5 4 3 2 1
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 0 1 0 2 -1 0 3 -2 1 0 4 -3 2 -1 0 5 -4
A036799 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 0 2 10 34 98 258 642 1538 3586 8194
A053088 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 0 1 0 3 -2 9 -12 31 -54 117 -224 459
A055087 \(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 0 0 0 1 0 1 0 1 2 0 1 2 0 1 2 3 0 1 2 3
A055642 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
A059100 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 3 6 11 18 27 38 51 66 83 102 123 146
A062805 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 0 1 4 18 112 975 11196 160132 2739136
A062806 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 0 1 10 102 1252 18555 324726 6565468
A067389 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 6 34 102 228 430 726 1134 1672 2358
A082375 \(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 0 1 2 0 3 1 4 2 0 5 3 1 6 4 2 0 7 5 3 1
A094053 \(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 0 1 0 2 2 0 3 4 3 0 4 6 6 4 0 5 8 9 8 5
A110660 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 0 0 2 2 6 6 12 12 20 20 30 30 42 42 56
A112367 \(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 0 0 1 0 1 3 0 1 3 6 0 1 3 6 10 0 1 3 6
A133819 \(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 1 4 1 4 9 1 4 9 16 1 4 9 16 25 1 4 9
A140960 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 0 -2 6 -18 46 -114 270 -626 1422 -3186
A141418 \(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 0 1 1 2 3 3 3 5 6 6 4 7 9 10 10 5 9 12
A167194 \((T_{n + 1, n - k + 1})^{-1}\) ➤ Tinvrev11 ➤ 1 -2 1 1 -2 1 0 1 -2 1 0 0 1 -2 1 0 0 0
A289399 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 0 3 21 102 426 1641 6015 21324 73812