Moebius
A363914 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ M^{-1}(n, k); M(n, k) = [k \le n \ \& \ k | n] } \)
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
---------- \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 2 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
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 -1 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
A000010 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 -1 -2 -2 -4 -2 -6 -4 -6 -4 -10 -4
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 1 2 2 4 2 6 4 6 4 10 4 12 6 8 8 16
---------- \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 1 2 2 4 2 6 4 6 4 10 4 12 6 8 8 16
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 1 2 2 4 2 6 4 6 4 10 4 12 6 8 8 16
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 -1 -2 -2 -4 -2 -6 -4 -6 -4 -10 -4
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 -1 -2 -2 -4 -2 -6 -4 -6 -4 -10 -4
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 -1 -2 -2 -4 -2 -6 -4 -6 -4 -10 -4
A000012 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 0 1 -1 0 1 0 -1 0 1 0 -1 0 0 1 0 0
---------- \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 -1 1 0 -1 1 0 -1 0 1 0 0 0 -1 1 0 0
---------- \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 0 -1 0 -1 0 0 0 -1 0 1 0 -1 0 0 0 0 0
---------- \(T_{n , n}\) ➤ 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
---------- \(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 0 1 -1 1 0 0 1 0 -1 1 0 -1 0 1 0
---------- \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ 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 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11
A000035 \(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 -1 0 -1 0 -1 0 -1 0 -1 0 -1 0 -1 0
A002378 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 0 2 6 12 20 30 42 56 72 90 110 132
A005563 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 0 -3 -8 -15 -24 -35 -48 -63 -80 -99
A007434 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 3 8 12 24 24 48 48 72 72 120 96 168
A007531 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 0 6 24 60 120 210 336 504 720 990
A008966 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 -1 -1 0 -1 1 -1 0 0 1 -1 0 -1 1 1 0
A027375 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 2 6 12 30 54 126 240 504 990 2046
A034444 \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 2 2 2 2 4 2 2 2 4 2 4 2 4 4 2 2 4 2
A051731 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 1 1 1 0 1 1 1 0 1 1 0 0 0 1 1 1 1 0 0
A054718 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 6 24 72 240 696 2184 6480 19656
A098018 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 -1 0 0 -1 1 -1 -1 1 1 -3 0 1 0 0
A113704 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 1 0 0 0
A113998 \(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 1 1 0 1 1 0 1 1 1 0 0 0 1 1 0 0 1 1
A127448 \(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 -1 2 -1 0 3 0 -2 0 4 -1 0 0 0 5 1 -2
A128174 \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 -1 -1 -1 0 -1 0 -1 0 -1 0 -1 0 -1 0
A174852 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 -1 1 -1 0 1 0 -1 0 1 -1 0 0 0 1 1 -1
A252764 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 2 24 240 3120 46410 823536 16773120
A363914 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 -1 1 0 -1 0 1 0 0 -1 0 1 0 -1 0
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 1 1 0 1 0 -1 0 0 -1 0 1 0 1 0
A365807 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 0 0 -1 0 0 -1 0 0 0 0 0 -1 0 0 1 0 0
A367773 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 3 -3 -3 -15 -39 -63 -15 -63 -735
A367774 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 -1 -3 -3 -15 9 -63 -15 -63 225