Catalan
A128899 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \sum_{i=1}^{n-k+1} \text{Catalan}(i) T_{k-1, n-i} } \)
\(T_{n, n - k}\) ➤ Trev ➤ 1 1 0 1 2 0 1 4 5 0 1 6 14 14 0 1 8 27
\(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 -2 0 1 -4 3 0 1 -6 10 -4 0 1 -8
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 0 0 1 0 2 0 5 1 0 14 4 0 42 14 1 0
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 0 1 0 2 3 0 5 9 10 0 14 28 34 35 0 42
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 2 2 5 8 3 14 28 18 4 42 96 81 32 5
\(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 14 48 110 208 350 544 798 1120 1518
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 2 20 42 3024 660 34320 450450
\(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 2 5 14 48 165 572 2002 7072 25194
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 2 5 14 48 110 429 910 3808 7752
\(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 5 48 429 3808 33915 303600 2731365
\(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 1 -3 4 5 -39 84 8 -603 1795 -1243
\(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 10 34 78 148 250 390 574 808 1098
\(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 8 78 920 12910 211932 4005372
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 0 2 0 4 5 0 6 14 14 0 8 27 48 42 0 10
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 0 0 2 0 5 4 0 14 14 6 0 42 48 27 8 0
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 0 1 2 1 4 0 1 6 5 1 8 14 0 1 10
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 1 1 3 3 1 5 10 10 1 7 21 35 35 1 9
\(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 0 1 -2 0 1 -4 5 0 1 -6 14 -14 0 1
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 0 2 0 4 10 0 6 28 42 0 8 54 144 168 0
\(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 1 6 15 70 210 924 3003 12870 43758
\(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 2 4 20 56 252 792 3432 11440 48620
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 2 4 14 27 110 208 910 1700 7752
\(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 0 -2 -4 -12 -36 -114 -372 -1244
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 0 2 24 188 1220 7116 38752 201176
\(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 5 58 1145 32966 1265677 61189668