Eulerian
A173018 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \sum_{j=0}^{k} (-1)^{j} \binom{n+1}{j} (k+1-j)^{n} } \)
A000007 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
A000012 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n , n}\) ➤ TablDiag0 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ RevPolyRow1 ➤ 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 ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A000142 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 2 6 24 120 720 5040 40320 362880
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 2 6 24 120 720 5040 40320 362880
A000295 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 4 11 26 57 120 247 502 1013 2036 4083
---------- \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 1 4 11 26 57 120 247 502 1013 2036
A000460 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 11 66 302 1191 4293 14608 47840
---------- \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 1 11 66 302 1191 4293 14608 47840
A000498 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 1 26 302 2416 15619 88234 455192
A000629 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 6 26 150 1082 9366 94586 1091670
A000670 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 3 13 75 541 4683 47293 545835
A000800 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 1 2 5 13 38 125 449 1742 7269
---------- \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 2 5 13 38 125 449 1742 7269 32433
A001286 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 1 6 36 240 1800 15120 141120
A001710 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 3 12 60 360 2520 20160 181440
---------- \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 3 12 60 360 2520 20160 181440
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 3 12 60 360 2520 20160 181440
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 3 12 60 360 2520 20160 181440
A002378 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 2 6 12 20 30 42 56 72 90 110 132 156
A006551 \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 1 4 11 66 302 2416 15619 156190
---------- \(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 1 4 11 66 302 2416 15619 156190
A008292 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 1 1 1 4 1 1 11 11 1 1 26 66 26 1 1 57
---------- \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 1 1 4 1 1 11 11 1 1 26 66 26 1 1 57
A009006 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 0 2 0 -16 0 272 0 -7936 0 353792 0
A011818 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 3 16 115 1056 11774 154624 2337507
A025585 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 4 66 2416 156190 15724248 2275172004
A028872 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 6 13 22 33 46 61 78 97 118 141 166
A038720 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 5 18 84 480 3240 25200 221760
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 5 18 84 480 3240 25200 221760
A065826 \(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 1 2 1 8 3 1 22 33 4 1 52 198 104 5 1
A122020 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 6 66 1140 28280 948570 41173776
A122704 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 4 22 160 1456 15904 202672 2951680
A122778 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 3 22 285 5656 158095 5881968
A123227 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 12 66 480 4368 47712 608016 8855040
A128103 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 1 2 12 68 360 2384 20160 185408
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 1 2 12 68 360 2384 20160 185408
A162498 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -1 1 3 -4 1 -23 33 -11 1 425 -620 220
---------- \((T_{n + 1, n - k + 1})^{-1}\) ➤ Tinvrev11 ➤ 1 -1 1 3 -4 1 -23 33 -11 1 425 -620 220
A173018 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 1 1 0 1 4 1 0 1 11 11 1 0 1 26
---------- \(T_{n, n - k}\) ➤ Trev ➤ 1 1 0 1 1 0 1 4 1 0 1 11 11 1 0 1 26 66
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 -1 1 0 -1 4 -1 0 -1 11 -11 1 0
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 0 1 -1 0 1 -4 1 0 1 -11 11 -1 0 1
A179929 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 2 6 -30 -42 882 -954 -39870 203958
A180056 \(T_{2 n, n}\) ➤ CentralE ➤ 1 1 11 302 15619 1310354 162512286
A180057 \(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 1 4 11 858 17214 14387280
A212846 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 -1 -3 15 21 -441 477 19935 -101979
A262745 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 1 4 12 52 360 2656 20160 177472
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 1 4 12 52 360 2656 20160 177472
A344052 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 1 -1 -8 19 276 -1002 -21216 103395
A344054 \(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 0 1 8 64 540 4920 48720 524160
A344393 \(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 0 1 1 1 4 0 1 11 1 1 26 11 0 1 57