Catalan
A128899 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \sum_{i=1}^{n-k+1} \text{Catalan}(i) T_{k-1, n-i} } \)
A000007 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
A000012 \(T_{n , n}\) ➤ TablDiag0 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ RevPolyRow1 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A000027 \(\sum_{k=0}^{1} T_{1, k}\ n^k\) ➤ PolyRow1 ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A000108 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 2 5 14 42 132 429 1430 4862 16796
---------- \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 -1 -2 -5 -14 -42 -132 -429 -1430
A000302 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 4 16 64 256 1024 4096 16384 65536
A000957 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 2 6 18 57 186 622 2120 7338 25724
A000984 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 2 6 20 70 252 924 3432 12870 48620
A001700 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 3 10 35 126 462 1716 6435 24310
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 3 10 35 126 462 1716 6435 24310
A001791 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 1 4 15 56 210 792 3003 11440 43758
A002057 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 4 14 48 165 572 2002 7072 25194 90440
A002457 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 6 30 140 630 2772 12012 51480
A003517 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 6 27 110 429 1638 6188 23256 87210
A005408 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
A005563 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 3 8 15 24 35 48 63 80 99 120 143 168
A005843 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
A014106 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 5 14 27 44 65 90 119 152 189 230 275
A025174 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 5 28 165 1001 6188 38760 245157
A026005 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 4 27 208 1700 14364 123970 1085760
A039598 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 2 1 5 4 1 14 14 6 1 42 48 27 8 1 132
A050166 \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 2 1 4 5 1 6 14 14 1 8 27 48 42 1 10
A064062 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 -3 13 -67 381 -2307 14589 -95235
A067336 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 8 34 148 652 2892 12882 57540
A078812 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -2 1 3 -4 1 -4 10 -6 1 5 -20 21 -8 1
A079273 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 10 29 58 97 146 205 274 353 442 541
A104530 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 15 78 411 2178 11574 61596 328083
A114121 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 7 26 99 382 1486 5812 22819 89846
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 7 26 99 382 1486 5812 22819 89846
A128899 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 2 1 0 5 4 1 0 14 14 6 1 0 42 48
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 -2 1 0 -5 4 -1 0 -14 14 -6 1 0
A128908 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 0 -2 1 0 3 -4 1 0 -4 10 -6 1 0 5
A172060 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 2 14 76 374 1748 7916 35096 153254
A172431 \(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -2 1 -4 3 1 -6 10 -4 1 -8 21 -20 5
A194723 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 5 29 181 1181 7941 54573 381333
A194724 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 7 58 523 4966 48838 492724 5068915
A224747 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 3 5 12 23 52 105 232 480 1049
A296770 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 5 24 111 500 2210 9632 41531 177564
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 5 24 111 500 2210 9632 41531 177564
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 5 24 111 500 2210 9632 41531 177564
A318829 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 2 1 2 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1
A359108 \(T_{2 n, n}\) ➤ CentralE ➤ 1 2 14 110 910 7752 67298 592020