Dyckpaths
A039599 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{2n}{n - k} (2k + 1) / (n + k + 1) } \)
A000007 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 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
---------- \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 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 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
---------- \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ RevPolyRow1 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A000108 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 1 2 5 14 42 132 429 1430 4862 16796
A000245 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 3 9 28 90 297 1001 3432 11934 41990
A000344 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 5 20 75 275 1001 3640 13260 48450
A000346 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 5 22 93 386 1586 6476 26333 106762
A000384 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 6 15 28 45 66 91 120 153 190 231 276
A000531 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 7 38 187 874 3958 17548 76627
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 1 7 38 187 874 3958 17548 76627
A000588 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 7 35 154 637 2548 9996 38760 149226
A000957 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -1 0 -1 -2 -6 -18 -57 -186 -622 -2120
A000958 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 1 3 8 24 75 243 808 2742 9458 33062
A000984 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 2 6 20 70 252 924 3432 12870 48620
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 2 6 20 70 252 924 3432 12870 48620
A001700 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 1 3 10 35 126 462 1716 6435 24310
---------- \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 3 10 35 126 462 1716 6435 24310
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 3 10 35 126 462 1716 6435 24310
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 1 3 10 35 126 462 1716 6435 24310
A002378 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 2 6 12 20 30 42 56 72 90 110 132 156
A005408 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
A007854 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 3 12 51 222 978 4338 19323 86310
A014107 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 2 9 20 35 54 77 104 135 170 209 252 299
A027849 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 20 87 232 485 876 1435 2192 3177 4420
A032443 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 3 11 42 163 638 2510 9908 39203
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 3 11 42 163 638 2510 9908 39203
A035610 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 4 28 232 2092 19864 195352 1970896
A039599 \(T_{n, k}\) ➤ Triangle ➤ 1 1 1 2 3 1 5 9 5 1 14 28 20 7 1 42 90
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 1 -1 2 -3 1 5 -9 5 -1 14 -28 20 -7 1
A050155 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 3 1 9 5 1 28 20 7 1 90 75 35 9 1 297
A050157 \(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 1 2 2 5 6 5 14 19 20 14 42 62 69 70
A050165 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 1 1 3 2 1 5 9 5 1 7 20 28 14 1 9 35
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -1 1 -3 2 1 -5 9 -5 1 -7 20 -28 14
A054142 \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 -1 1 -3 1 1 -5 6 -1 1 -7 15 -10 1 1
A062158 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 5 20 51 104 185 300 455 656 909 1220
A062344 \(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 2 1 4 6 1 6 15 20 1 8 28 56 70 1 10
A064062 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -1 3 -13 67 -381 2307 -14589 95235
A076035 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 4 20 104 548 2904 15432 82128 437444
A085478 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 -1 1 1 -3 1 -1 6 -5 1 1 -10 15 -7 1
A089022 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 3 15 87 543 3543 23823 163719 1143999
A126596 \(T_{2 n, n}\) ➤ CentralE ➤ 1 3 20 154 1260 10659 92092 807300
A174687 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 2 9 48 275 1638 9996 62016 389367
A210551 \(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -3 1 -5 6 1 -7 15 -10 1 -9 28 -35
A258993 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -3 1 6 -5 1 -10 15 -7 1 15 -35 28 -9
A274115 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 2 4 8 17 35 75 157 337 712 1529
A296771 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 3 13 58 257 1126 4882 20980 89497
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 3 13 58 257 1126 4882 20980 89497
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 3 13 58 257 1126 4882 20980 89497