Binomialcatalan
A124644 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k} \text{Catalan}(n - k) } \)
A000012 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A000027 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
---------- \(\sum_{k=0}^{1} T_{1, k}\ n^k\) ➤ 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
A000034 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
A000108 \(T_{n , n}\) ➤ TablDiag0 ➤ 1 1 2 5 14 42 132 429 1430 4862 16796
A000888 \(T_{2 n, n}\) ➤ CentralE ➤ 1 2 12 100 980 10584 121968 1472328
A000891 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 1 3 20 175 1764 19404 226512 2760615
A000984 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 1 2 6 20 70 252 924 3432 12870 48620
A001405 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -1 2 -3 6 -10 20 -35 70 -126 252 -462
A001844 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 1 5 13 25 41 61 85 113 145 181 221 265
A002378 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 2 6 12 20 30 42 56 72 90 110 132 156
A002522 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 2 5 10 17 26 37 50 65 82 101 122 145
A005043 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 0 1 -1 3 -6 15 -36 91 -232 603 -1585
A005491 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 5 15 37 77 141 235 365 537 757 1031
A007317 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 2 5 15 51 188 731 2950 12235 51822
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 2 5 15 51 188 731 2950 12235 51822
A026375 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 3 11 45 195 873 3989 18483 86515
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 3 11 45 195 873 3989 18483 86515
A026376 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 6 30 144 685 3258 15533 74280
A064613 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 3 10 37 150 654 3012 14445 71398
A090344 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 1 2 3 6 11 23 47 102 221 493 1105
A092443 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 1 3 12 50 210 882 3696 15444 64350
A098465 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 1 3 7 27 91 373 1457 6163 25795
A098474 \(T_{n, k}\) ➤ Triangle ➤ 1 1 1 1 2 2 1 3 6 5 1 4 12 20 14 1 5 20
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 1 -1 1 -2 2 1 -3 6 -5 1 -4 12 -20 14
A102318 \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 3 8 27 97 373 1493 6163 26027
A104455 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 4 17 77 371 1890 10095 56040 320795
A105864 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 3 7 21 65 215 735 2585 9281 33883
A124644 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 1 2 2 1 5 6 3 1 14 20 12 4 1 42 70
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -1 2 -2 1 5 -6 3 -1 14 -20 12 -4 1
A125558 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 6 50 490 5292 60984 736164 9202050
A134481 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 5 20 50 100 175 280 420 600 825 1100
A162326 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 3 13 71 441 2955 20805 151695 1135345
A292632 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 2 10 77 798 10392 162996 2991340
A337167 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 4 25 199 1795 17422 177463 1870960
A337168 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -1 5 -21 105 -553 3053 -17405 101713
A338979 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 2 13 199 5073 181776 8413021