Motzkin
A064189 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k} \text{Hyper}([(k-n)/2, (k-n+1)/2], [k+2], 4) } \)
\(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 -1 1 -2 0 1 -3 1 1 1 -4 3 2 -1 1 -5
\(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 2 1 5 3 1 12 9 4 1 30 25 14 5 1 76 69
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 2 1 3 5 1 4 9 12 1 5 14 25 30 1 6
\(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -2 1 1 -3 1 2 3 -4 1 -4 2 6 -5 1 2 -9
\(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -2 1 -3 1 1 -4 3 2 1 -5 6 2 -4 1 -6
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 1 2 2 4 5 4 9 12 13 9 21 30 34 35 21
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 2 2 5 6 3 12 18 12 4 30 50 42 20 5 76
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 2 60 36 1050 4903140 63847980
\(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 2 10 73 713 8796 131727 2325324
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 1 2 2 3 5 4 4 9 12 9 5 14 25 30 21 6 20
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 1 2 2 4 5 3 9 12 9 4 21 30 25 14 5 51
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 1 1 2 1 3 2 1 4 5 1 5 9 4 1 6 14
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 2 1 3 5 1 4 9 13 1 5 14 26 35 1 6
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 1 2 4 3 10 12 4 18 36 36 5 28 75 120
\(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 3 6 19 45 141 357 1107 2907 8953
\(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 1 2 7 16 51 126 393 1016 3139 8350
\(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 1 6 25 94 333 1140 3813 12546 40777
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 1 10 59 292 1291 5322 20877 78972
\(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 13 59 163 349 641 1063 1639 2393 3349
\(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 4 25 163 1147 8350 62623 479488
\(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 2 13 163 3233 87876 3070117 131170404