Eulersec
A119879 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k}\, 2^{n-k}\ \text{Euler} (n-k, 1/2) } \)
\(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 0 1 -3 0 1 0 -6 0 1 25 0 -10 0 1 0 75
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 0 1 0 -3 1 0 -6 0 1 0 -10 0 25 1 0
\(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 0 1 3 0 1 0 6 0 1 5 0 10 0 1 0 15 0
\(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 0 1 0 3 1 0 6 0 1 0 10 0 5 1 0 15 0
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 0 -1 1 0 0 5 -3 1 0 0 0 -61 25 -6 1 0
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 0 1 -1 -1 0 0 -3 -3 -2 5 5 -1 -1 0 0
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 0 2 -3 0 3 0 -12 0 4 25 0 -30 0 5 0
\(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 0 -6 0 75 0 -1708 0 62325 0 -3334386
\(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 0 -10 0 175 0 -5124 0 228525 0
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 1 3 30 50 4575 32025 11827900
\(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 1 3 6 25 75 427 1708 12465 62325
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 -2 -8 8 96 -96 -2176 2176 79360
\(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 0 0 3 0 -41 0 1024 0 -39603 0
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 0 -3 -6 0 0 175 350 0 0 -28182
\(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 -3 0 175 0 -28182 0 8912475 0
\(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 4 6 -8 -40 96 672 -2176 -19584
\(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 0 -8 -30 26 840 2696 -22722 -240146
\(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 1 0 -8 -30 26 840 2696 -22722 -240146
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 0 0 -1 0 -3 0 0 -6 0 5 0 -10 0 25 0 0
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 0 -1 0 0 -3 0 5 0 -6 0 0 25 0 -10 0 -61
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 0 1 0 1 0 -1 1 0 -3 1 0 -6 0 1 0
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 1 1 1 0 1 1 -2 -2 1 1 -5 -5 0 1 1
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 0 0 -2 0 -6 0 0 -12 0 20 0 -20 0 100 0
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 -2 -8 8 96 -96 -2176 2176 79360
\(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 1 0 -2 -5 -9 -9 5 48 140 245 149
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 0 0 -6 -10 0 0 350 630 0 0 -56364
\(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 0 -10 0 630 0 -104676 0 33669350 0
\(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 -2 -6 8 80 -96 -1904 2176 71424
\(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 -2 -8 8 96 -96 -2176 2176 79360
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 0 -4 -12 56 360 -1056 -12656 32640