Lah
A271703 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k} \text{FallingFactorial}(n-1, n-k) } \)
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 \(T_{n , n}\) ➤ 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
A000142 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 2 6 24 120 720 5040 40320 362880
A000262 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 3 13 73 501 4051 37633 394353
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 3 13 73 501 4051 37633 394353
A001053 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 2 7 30 157 972 6961 56660 516901
A001286 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 6 36 240 1800 15120 141120 1451520
A001754 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 12 120 1200 12600 141120 1693440
A002378 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 2 6 12 20 30 42 56 72 90 110 132 156
---------- \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 2 6 12 20 30 42 56 72 90 110 132
A002720 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 7 34 209 1546 13327 130922 1441729
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 7 34 209 1546 13327 130922 1441729
A002868 \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 2 6 36 240 1800 15120 141120
A003154 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 13 37 73 121 181 253 337 433 541 661
A005408 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
A005563 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 3 8 15 24 35 48 63 80 99 120 143 168
A025168 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 5 37 361 4361 62701 1044205
A052852 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 4 21 136 1045 9276 93289 1047376
A062147 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 5 31 229 1961 19081 207775 2501801
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 5 31 229 1961 19081 207775 2501801
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 5 31 229 1961 19081 207775 2501801
A083374 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 6 36 120 300 630 1176 2016 3240 4950
A088312 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 1 6 37 260 2101 19362 201097
A088313 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 2 7 36 241 1950 18271 193256
A089231 \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 2 1 6 6 1 12 36 24 1 20 120 240 120
---------- \(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -2 1 -6 6 1 -12 36 -24 1 -20 120
A096939 \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 2 6 36 260 1950 19362 193256
A096965 \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 1 7 37 241 2101 18271 201097
A103194 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 6 39 292 2505 24306 263431 3154824
A105278 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 2 1 6 6 1 24 36 12 1 120 240 120 20 1
---------- \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -2 1 6 -6 1 -24 36 -12 1 120 -240 120
A111884 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 -1 -1 1 19 151 1091 7841 56519
A180047 \(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 0 0 1 0 2 0 6 1 0 24 6 0 120 36 1 0
A187535 \(T_{2 n, n}\) ➤ CentralE ➤ 1 2 36 1200 58800 3810240 307359360
A226514 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 13 44 99 184 305 468 679 944 1269
A248045 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 6 120 4200 211680 13970880 1141620480
A253285 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 24 240 1200 4200 11760 28224 60480
A255806 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 15 99 801 7623 83079 1017495
A271703 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 2 1 0 6 6 1 0 24 36 12 1 0 120
---------- \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 0 -2 1 0 6 -6 1 0 -24 36 -12 1 0
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 -2 1 0 -6 6 -1 0 -24 36 -12 1
A293145 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 8 99 1696 37225 997056 31535371
A293146 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 5 73 2161 108101 8201701 878797165
A317364 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 0 4 16 48 64 -800 -12288 -127232
A318223 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 -3 13 -71 441 -2699 9157 206193
A321837 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 7 73 1009 17341 355951 8488117
A343581 \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 2 6 36 240 1200 12600 58800 846720
A344050 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 1 -3 1 73 -699 3001 24697 -783999
A344051 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 5 37 361 4301 60001 954325 16984577
A349776 \(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 0 1 0 2 3 0 6 12 13 0 24 60 72 73 0
A359365 \(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 2 6 72 240 3600 75600 1411200
A367776 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 6 240 12600 846720 69854400