Schroederp
A104684 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k} \binom{2n - k}{n} } \)
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
---------- \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 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 ➤ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
A000984 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 2 6 20 70 252 924 3432 12870 48620
A001850 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 3 13 63 321 1683 8989 48639 265729
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 3 13 63 321 1683 8989 48639 265729
---------- \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -3 13 -63 321 -1683 8989 -48639
A002378 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 2 6 12 20 30 42 56 72 90 110 132 156
A002426 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 3 7 19 51 141 393 1107 3139 8953
A002457 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 6 30 140 630 2772 12012 51480 218790
A002544 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 12 90 560 3150 16632 84084 411840
A003154 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 13 37 73 121 181 253 337 433 541 661
A005258 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 3 19 147 1251 11253 104959 1004307
A005408 \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ RevPolyRow1 ➤ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
A006442 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 5 37 305 2641 23525 213445 1961825
A006480 \(T_{2 n, n}\) ➤ CentralE ➤ 1 6 90 1680 34650 756756 17153136
A028872 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 6 13 22 33 46 61 78 97 118 141 166 193
A033487 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 6 30 90 210 420 756 1260 1980 2970 4290
A047665 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 6 31 160 841 4494 24319 132864
A063007 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 2 1 6 6 1 12 30 20 1 20 90 140 70 1
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -2 1 -6 6 1 -12 30 -20 1 -20 90
A069835 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 4 22 136 886 5944 40636 281488
A084768 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 7 73 847 10321 129367 1651609
A084771 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 5 33 245 1921 15525 127905 1067925
A104684 \(T_{n, k}\) ➤ Triangle ➤ 1 2 1 6 6 1 20 30 12 1 70 140 90 20 1
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 2 -1 6 -6 1 20 -30 12 -1 70 -140 90
A108666 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 8 57 384 2505 16008 100849 628736
A126869 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 0 -2 0 6 0 -20 0 70 0 -252 0 924 0
A160674 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 63 305 847 1809 3311 5473 8415 12257
A208881 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 2 30 560 11550 252252 5717712 133024320
A226994 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 2 7 32 161 842 4495 24320 132865
A331656 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 3 37 847 28401 1256651 69125869
A335309 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 3 22 245 3606 65527 1411404 35066313
A349713 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 2 7 26 101 404 1645 6784 28243 118442