Narayana
A090181 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{n-k} \binom{n-1}{n-k} \frac{1}{n-k+1} } \)
\(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 0 -1 1 0 2 -3 1 0 -7 12 -6 1 0 39
\(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 -1 0 1 -3 2 0 1 -6 12 -7 0 1
\(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -1 1 -3 2 1 -6 12 -7 1 -10 40 -70
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 0 0 1 0 1 0 1 1 0 1 3 0 1 6 1 0 1 10
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 1 3 6 20 150 525 980 7056 52920
\(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 1 3 6 10 5 7 14 12 9 11 11 13 13 1
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 1 1 6 10 50 105 490 1176 5292 13860
\(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 3 13 65 356 2072 12601 79221 511174
\(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 1 -1 -5 9 56 -120 -825 1925 14014
\(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 6 57 740 12130 239442 5516133
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 0 1 0 3 1 0 6 6 1 0 10 20 10 1 0 15 50
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 0 0 1 0 1 3 0 1 6 6 0 1 10 20 10 0 1 15
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 0 1 1 1 3 0 1 6 1 1 10 6 0 1 15
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 1 1 2 2 1 4 5 5 1 7 13 14 14 1 11
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 0 1 0 3 2 0 6 12 3 0 10 40 30 4 0 15