Schroeder
A122538 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ is(k = 0)\ ? \ 0^{n} : T(n-1,k-1)+T(n-1,k)+T(n,k+1) } \)
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 0 0 1 0 2 0 6 1 0 22 4 0 90 16 1 0
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 0 1 0 2 3 0 6 10 11 0 22 38 44 45 0
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 2 2 6 8 3 22 32 18 4 90 136 90 32 5
\(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 6 30 146 714 3534 17718 89898 461010
\(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 22 68 146 264 430 652 938 1296 1734
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 2 12 528 6120 65577360 2384317320
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 5 27 149 829 4633 25975 146009
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 2 6 16 68 146 714 1408 7432 14002
\(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 6 68 714 7432 77550 812940 8561938
\(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 11 36 81 152 255 396 581 816 1107
\(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 15 81 453 2583 14907 86733 507561
\(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 8 81 984 14025 231468 4372417
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 0 2 0 4 6 0 6 16 22 0 8 30 68 90 0 10
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 0 0 2 0 6 4 0 22 16 6 0 90 68 30 8 0
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 0 1 2 1 4 0 1 6 6 1 8 16 0 1 10
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 0 2 0 4 12 0 6 32 66 0 8 60 204 360 0
\(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 2 4 28 76 550 1688 12536 41128
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 5 27 149 829 4633 25975 146009
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 2 4 16 30 146 264 1408 2490 14002
\(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 0 -4 -12 -52 -228 -1052 -5004
\(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 2 16 104 632 3730 21696 125216
\(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 5 27 149 829 4633 25975 146009
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 0 2 28 268 2180 16230 114408 777560
\(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 7 67 757 9421 124771 1725319
\(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 5 67 1689 65541 3491053 238013959