Schroederp
A104684 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k} \binom{2n - k}{n} } \)
\(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 -2 1 6 -6 1 -32 42 -12 1 310 -440 150
\(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 -2 1 -6 6 1 -12 42 -32 1 -20 150
\(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 6 1 30 12 1 140 90 20 1 630 560 210
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 6 1 12 30 1 20 90 140 1 30 210 560
\(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -6 1 42 -12 1 -440 150 -20 1 7110
\(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -6 1 -12 42 1 -20 150 -440 1 -30
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 2 6 1 20 6 70 30 1 252 140 12 924 630
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 2 3 6 12 13 20 50 62 63 70 210 300
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 6 2 30 24 3 140 180 60 4 630 1120 630
\(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 20 210 1680 11550 72072 420420
\(T_{n + 3, n}\) ➤ TablDiag3 ➤ 20 140 560 1680 4200 9240 18480 34320
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 2 6 60 1260 5040 277200 5405400
\(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 2 6 2 10 2 42 2 6 2 22 2 26 2 2 2 34
\(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 2 6 30 140 630 3150 16632 84084
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 5 31 195 1221 7593 46915 288263
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 4 21 120 705 4188 24997 149488 894465
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 2 6 30 90 560 1680 11550 34650 252252
\(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 4 21 120 705 4188 24997 149488 894465
\(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 10 87 696 5265 38298 270655 1870816
\(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 -1 -5 35 -29 -751 3991 4115 -137885
\(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 20 63 136 245 396 595 848 1161 1540
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 2 6 6 12 30 20 20 90 140 70 30 210 560
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 2 6 6 20 30 12 70 140 90 20 252 630 560
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 2 1 6 1 12 6 1 20 30 1 30 90 20 1
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 3 1 7 13 1 13 43 63 1 21 111 251
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 2 6 12 12 60 60 20 180 420 280 30 420
\(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 7 31 161 841 4495 24319 132865
\(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 2 6 32 160 842 4494 24320 132864
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 5 31 195 1221 7593 46915 288263
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 6 12 90 210 1680 4200 34650 90090
\(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 12 210 4200 90090 2018016 46558512
\(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 2 18 132 900 5910 37926 239624
\(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 5 31 195 1221 7593 46915 288263
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 2 30 312 2760 22290 169806 1242080