Fallingfact
A008279 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ n! / (n - k)! } \)
A000012 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A000023 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -1 2 -2 8 8 112 656 5504 49024 491264
A000027 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
---------- \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(\sum_{k=0}^{1} T_{1, k}\ n^k\) ➤ PolyRow1 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
---------- \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ RevPolyRow1 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A000142 \(T_{n , n}\) ➤ TablDiag0 ➤ 1 1 2 6 24 120 720 5040 40320 362880
---------- \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 1 2 6 24 120 720 5040 40320 362880
---------- \(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 2 6 24 120 720 5040 40320 362880
---------- \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 2 6 24 120 720 5040 40320 362880
A000166 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 0 1 -2 9 -44 265 -1854 14833 -133496
A000354 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -1 5 -29 233 -2329 27949 -391285
A000407 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 6 60 840 15120 332640 8648640
A000522 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 2 5 16 65 326 1957 13700 109601
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 2 5 16 65 326 1957 13700 109601
A001339 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 3 11 49 261 1631 11743 95901 876809
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 3 11 49 261 1631 11743 95901 876809
A001710 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 1 3 12 60 360 2520 20160 181440 1814400
A001715 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 1 4 20 120 840 6720 60480 604800
A001813 \(T_{2 n, n}\) ➤ CentralE ➤ 1 2 12 120 1680 30240 665280 17297280
A001844 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 1 5 13 25 41 61 85 113 145 181 221 265
A002378 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 2 6 12 20 30 42 56 72 90 110 132 156
A002522 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 2 5 10 17 26 37 50 65 82 101 122 145
A002720 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 2 7 34 209 1546 13327 130922 1441729
A002747 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 2 9 28 185 846 7777 47384 559953
A003470 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 3 8 31 147 853 5824 45741 405845
A006963 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 1 3 20 210 3024 55440 1235520 32432400
A007526 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 1 4 15 64 325 1956 13699 109600
A007531 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 6 24 60 120 210 336 504 720 990 1320
A008279 \(T_{n, k}\) ➤ Triangle ➤ 1 1 1 1 2 2 1 3 6 6 1 4 12 24 24 1 5 20
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 1 -1 1 -2 2 1 -3 6 -6 1 -4 12 -24 24
A009179 \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 3 9 37 185 1111 7777 62217 559953
A009940 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 0 -1 -4 -15 -56 -185 -204 6209 112400
A010842 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 3 10 38 168 872 5296 37200 297856
A010844 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 3 13 79 633 6331 75973 1063623
A010845 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 4 25 226 2713 40696 732529 15383110
A030297 \(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 1 6 27 124 645 3906 27391 219192
A053486 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 4 17 78 393 2208 13977 100026 806769
A063170 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 2 10 78 824 10970 176112 3309110
A068424 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 2 2 3 6 6 4 12 24 24 5 20 60 120 120
A072374 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 1 2 3 6 11 24 51 122 291 756 1979
A081125 \(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 2 6 12 60 120 840 1680 15120 30240
A087208 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 1 3 7 37 141 1111 5923 62217 426457
A093964 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 6 33 196 1305 9786 82201 767208
A094587 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 1 2 2 1 6 6 3 1 24 24 12 4 1 120
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -1 2 -2 1 6 -6 3 -1 24 -24 12 -4 1
A111063 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 3 9 31 129 651 3913 27399 219201
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 3 9 31 129 651 3913 27399 219201
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 3 9 31 129 651 3913 27399 219201
A128229 \((T_{n, n - k})^{-1}\) ➤ Tinvrev ➤ 1 -1 1 0 -2 1 0 0 -3 1 0 0 0 -4 1 0 0 0
A173333 \(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 1 2 1 6 3 1 24 12 4 1 120 60 20 5 1 720
A181511 \(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 1 1 2 1 3 6 1 4 12 24 1 5 20 60 120 1 6
A186763 \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 1 2 7 28 141 846 5923 47384 426457
A205825 \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 1 2 3 12 20 120 210 1680 3024 30240
A277452 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 2 13 226 7889 458026 39684637
A343276 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 10 81 652 5545 50886 506905 5480056
A344391 \(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 1 1 1 1 2 1 3 2 1 4 6 1 5 12 6 1 6 20
A347667 \(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 1 2 1 3 5 1 4 10 16 1 5 17 41 65 1 6
A367962 \(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 2 2 4 5 6 12 15 16 24 48 60 64 65