Lah
A271703 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k} \text{FallingFactorial}(n-1, n-k) } \)
\(T_{n, n - k}\) ➤ Trev ➤ 1 1 0 1 2 0 1 6 6 0 1 12 36 24 0 1 20
\(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 -2 0 1 -6 6 0 1 -12 36 -24 0 1
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 2 2 6 12 3 24 72 36 4 120 480 360 80
\(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 8 44 304 2512 24064 261536 3173888
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 0 2 0 6 6 0 12 36 24 0 20 120 240 120 0
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 0 0 2 0 6 6 0 24 36 12 0 120 240 120 20
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 0 1 2 1 6 0 1 12 6 1 20 36 0 1 30
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 1 1 3 3 1 7 13 13 1 13 49 73 73 1
\(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 0 1 -2 0 1 -6 6 0 1 -12 36 -24 0 1
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 0 2 0 6 12 0 12 72 72 0 20 240 720 480
\(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 3 7 19 57 175 583 2007 7249 27187
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 2 6 36 120 1200 4200 58800 211680
\(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 2 18 156 1460 15030 170142 2107448
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 0 2 30 372 4580 58830 801402 11635400