Binomialcatalan
A124644 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k} \text{Catalan}(n - k) } \)
\((T_{n, n - k})^{-1}\) ➤ Tinvrev ➤ 1 -1 1 0 -2 1 1 0 -3 1 2 4 0 -4 1 -2 10
\(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 2 2 3 6 5 4 12 20 14 5 20 50 70 42 6
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 2 2 5 6 3 14 20 12 4 42 70 50 20 5
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 1 1 1 1 2 1 3 2 1 4 6 1 5 12 5 1 6 20
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 1 2 1 3 5 1 4 10 15 1 5 17 37 51 1 6
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 2 4 3 12 15 4 24 60 56 5 40 150 280
\(T_{n + 3, n}\) ➤ TablDiag3 ➤ 1 4 20 100 490 2352 11088 51480 235950
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 2 30 420 2100 69300 6306300 1681680
\(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 2 6 20 70 252 924 3696 15444 64350
\(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 2 8 24 97 358 1493 6072 26027
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 3 9 30 111 443 1859 8067 35835 161937
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 1 2 3 12 20 100 175 980 1764 10584
\(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 10 72 456 2705 15438 85925 469976
\(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 2 7 33 183 1118 7281 49626 349999
\(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 0 -1 -5 -9 16 185 624 175 -9080
\(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 1 15 71 199 429 791 1315 2031 2969 4159
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 1 2 1 6 3 1 20 12 4 1 70 50 20 5 1 252
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 1 1 2 1 3 6 1 4 12 20 1 5 20 50 70 1 6
\(T^{-1}_{n + 1, n-k}\) ➤ RevTinv11 ➤ 1 -2 1 0 -3 1 4 0 -4 1 10 10 0 -5 1 -12
\(T^{-1}_{n + 1, n-k}\) ➤ RevTrevinv11 ➤ 1 1 -2 1 -3 0 1 -4 0 4 1 -5 0 10 10 1
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 2 1 5 2 14 6 1 42 20 3 132 70 12 1
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 2 2 4 5 5 11 14 15 14 34 46 50 51
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 1 2 2 6 6 3 20 24 12 4 70 100 60 20 5
\(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 1 2 7 24 91 358 1457 6072 25795
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 3 9 30 111 443 1859 8067 35835 161937
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 2 6 12 50 100 490 980 5292 10584
\(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 1 4 15 60 255 1128 5117 23600 110115
\(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 3 9 30 111 443 1859 8067 35835 161937
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 1 6 27 120 555 2658 13013 64536