Binarypell
A038207 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k} \, 2^{n-k} } \)
\(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 4 1 12 6 1 32 24 8 1 80 80 40 10 1
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 4 1 6 12 1 8 24 32 1 10 40 80 80 1
\(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -4 1 12 -6 1 -32 24 -8 1 80 -80 40
\(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -4 1 -6 12 1 -8 24 -32 1 -10 40 -80
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 2 3 4 8 9 8 20 26 27 16 48 72 80 81
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 4 2 12 12 3 32 48 24 4 80 160 120 40
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 2 4 24 96 160 960 13440 35840 32256
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 2 4 12 24 80 160 560 1120 4032 8064
\(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 8 45 216 945 3888 15309 58320
\(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 -1 -3 11 1 -81 141 363 -1791 479
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 2 4 4 8 12 6 16 32 24 8 32 80 80 40 10
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 3 1 5 9 1 7 19 27 1 9 33 65 81 1 11
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 2 4 8 6 24 24 8 48 96 64 10 80 240 320