Monotone
A059481 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n+k-1}{k} } \)
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 3 2 10 6 3 35 20 10 4 126 70 35 15 5
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 1 1 1 1 2 1 3 3 1 4 6 1 5 10 10 1 6
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 2 6 3 12 30 4 20 60 140 5 30 105 280
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 6 30 140 630 5544 60060 51480
\(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 1 4 7 46 86 610 1163 8518 16414
\(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 2 13 24 166 314 2269 4352 32206
\(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 14 117 784 4650 25542 133133 668096
\(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 1 20 111 334 749 1416 2395 3746 5529
\(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 4 34 334 3478 37384 409960 4558306
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 3 1 10 2 35 6 1 126 20 3 462 70 10
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 2 3 5 6 10 16 19 20 35 55 65 69 70
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 1 2 2 6 6 3 20 20 12 4 70 70 45 20 5
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 2 6 10 35 56 210 330 1287 2002 8008
\(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 6 35 210 1287 8008 50388 319770
\(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -1 9 -61 433 -3141 23145 -172509
\(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 10 20 42 82 146 240 370 542 762 1036