Partition
A072233 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ T_{n - 1, k - 1} + T_{n - k, k} } \)
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
A000009 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 2 2 3 4 5 6 8 10 12 15 18 22 27
A000012 \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 -1 0 1 -1 0 0 1 -1 -1 1 0 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
---------- \(T_{n , n}\) ➤ 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 ➤ 0 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
A000041 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 2 3 5 7 11 15 22 30 42 56 77 101
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 2 3 5 7 11 15 22 30 42 56 77 101
---------- \(T_{2 n, n}\) ➤ CentralE ➤ 1 1 2 3 5 7 11 15 22 30 42 56 77 101
---------- \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 1 2 3 5 7 11 15 22 30 42 56 77 101
A000065 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 1 2 4 6 10 14 21 29 41 55 76 100 134
A000700 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 0 -1 1 -1 1 -1 2 -2 2 -2 3 -3 3 -4
A000701 \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 1 1 2 3 5 7 10 14 20 27 37 49 66 86
A001399 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 1 2 3 4 5 7 8 10 12 14 16 19 21 24 27
A002061 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 3 7 13 21 31 43 57 73 91 111 133 157
A002378 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 2 6 12 20 30 42 56 72 90 110 132 156
A002569 \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 1 1 2 2 3 4 5 7 9 11 15 18 23 30 37
A002865 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 1 2 2 4 4 7 8 12 14 21 24 34 41
A004526 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10
A006128 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 3 6 12 20 35 54 86 128 192 275 399
A008284 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1
A010701 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 1 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
A027187 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 1 1 3 3 6 7 12 14 22 27 40 49 69 86
A027193 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 1 2 2 4 5 8 10 16 20 29 37 52 66 90
A027444 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 3 14 39 84 155 258 399 584 819 1110
A038498 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -1 1 0 -1 1 1 -1 -1 1 0 1 -1 -1 1 0 1
A046682 \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 1 2 3 4 6 8 12 16 22 29 40 52 69 90
A055642 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
A058397 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 3 6 13 22 42 66 112 172 270 397 602
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 3 6 13 22 42 66 112 172 270 397 602
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 3 6 13 22 42 66 112 172 270 397 602
A058398 \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 2 3 3
A066639 \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 1 1 2 2 3 4 5 6 7 10 11 14 15 21 22
A070933 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 6 14 34 74 166 350 746 1546 3206
A071109 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 2 -6 14 -26 50 -102 214 -426 834
A072233 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 1 1 0 1 1 1 0 1 2 1 1 0 1 2 2 1
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 -1 1 0 -1 1 -1 0 -1 2 -1 1 0
A075900 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 3 7 19 43 115 259 659 1523 3731
A093694 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 5 9 17 27 46 69 108 158 234 331 476
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 5 9 17 27 46 69 108 158 234 331 476
A098545 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 3 7 21 51 148 365 983 2461 6360
A119620 \(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 1 1 2 2 3 3 5 5 7 7 11 11 15 15 22
A124577 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 6 39 356 4055 57786 983535 19520264
A172467 \(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 1 2 1 2 3 1 4 3 4 1 4 6 4 5 1 6 9 8 5
A196087 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 1 3 8 15 31 51 90 142 228 341 525
A242587 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 12 39 129 399 1245 3783 11514 34734
A296010 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 5 14 34 68 133 232 402 652 1048
A300579 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 4 13 49 157 589 1885 6826 22378
A338697 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 3 13 101 931 12391 178809 3331721
A352402 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 -1 3 -1 7 -1 15 31 63 159 95 671
A373820 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1