Orderedcycle
A225479 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ k! {n \brack k} } \)
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 \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ 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 ➤ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
A000142 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 1 2 6 24 120 720 5040 40320 362880
---------- \(T_{n , n}\) ➤ TablDiag0 ➤ 1 1 2 6 24 120 720 5040 40320 362880
A001286 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 1 6 36 240 1800 15120 141120 1451520
A006252 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 1 -2 4 -14 38 -216 600 -6240 9552
A007840 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 3 14 88 694 6578 72792 920904
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 3 14 88 694 6578 72792 920904
A014105 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 3 10 21 36 55 78 105 136 171 210 253
A048594 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 1 2 2 6 6 6 22 36 24 24 100 210 240
A051890 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 6 14 26 42 62 86 114 146 182 222 266
A052517 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 2 6 22 100 548 3528 26136 219168
A052748 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 6 36 210 1350 9744 78792 708744 7036200
A052801 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 8 46 342 3108 33324 411360 5741856
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 8 46 342 3108 33324 411360 5741856
A052811 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 2 6 46 340 3308 36288 460752
A058583 \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 2 6 36 240 2040 21000 235200
A075181 \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 2 1 6 6 2 24 36 22 6 120 240 210 100
A075182 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 2 2 2 2 2 24 24 24 24 48 48 48 48
A088500 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 10 76 772 9808 149552 2660544
A088501 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 6 -28 172 -1328 12272 -132480
A129841 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 1 4 12 52 256 1502 10158 78360
A215916 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 5 32 254 2414 26746 338568 4820952
A225479 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 1 2 0 2 6 6 0 6 22 36 24 0 24
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 -1 2 0 -2 6 -6 0 -6 22 -36 24
A227917 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 4 26 232 2624 35888 575280 10569984
A238685 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 6 210 17640 2693880 649479600
A277759 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 4 30 324 4540 78060 1589448
A308565 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 1 0 -6 -12 140 1020 -5208 -117264
A317171 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 10 222 8824 553870 50545008
A344498 \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 1 2 22 100 1350 9744 162456 1614816
A352069 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 5 42 492 7374 134478 2887128
A352074 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 4 42 904 34070 2019888 174588120
A354237 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 0 2 -8 64 -592 6768 -90624 1395840
A354263 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 21 222 3132 55242 1169262 28873800
A376873 \(T_{2 n, n}\) ➤ CentralE ➤ 1 1 22 1350 162456 32319000 9604465200