Stirlingcycb
A028338 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \sum_{i=k}^{n} (-2)^{n-i} \binom{i}{k} {n \brack i} } \)
A000007 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 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
---------- \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 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 ➤ 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
A000165 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 2 8 48 384 3840 46080 645120 10321920
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 2 8 48 384 3840 46080 645120 10321920
A000290 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 1 4 9 16 25 36 49 64 81 100 121 144 169
A000567 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 8 21 40 65 96 133 176 225 280 341 408
A001147 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 1 3 15 105 945 10395 135135 2027025
---------- \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 3 15 105 945 10395 135135 2027025
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -1 -1 -3 -15 -105 -945 -10395 -135135
A002866 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 1 4 24 192 1920 23040 322560 5160960
---------- \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 4 24 192 1920 23040 322560 5160960
---------- \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 4 24 192 1920 23040 322560 5160960
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 4 24 192 1920 23040 322560 5160960
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 1 4 24 192 1920 23040 322560 5160960
A004041 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 4 23 176 1689 19524 264207 4098240
---------- \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 4 23 176 1689 19524 264207 4098240
A005563 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 3 8 15 24 35 48 63 80 99 120 143 168
A007696 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -1 5 -45 585 -9945 208845 -5221125
A008545 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 3 21 231 3465 65835 1514205 40883535
A024196 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 3 23 86 230 505 973 1708 2796 4335 6435
A024197 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 15 176 950 3480 10045 24640 53676
A028338 \(T_{n, k}\) ➤ Triangle ➤ 1 1 1 3 4 1 15 23 9 1 105 176 86 16 1
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 1 -1 3 -4 1 15 -23 9 -1 105 -176 86
A028339 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 9 86 950 12139 177331 2924172
A028340 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 16 230 3480 57379 1038016 20570444
A039755 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 -1 1 1 -4 1 -1 13 -9 1 1 -40 58 -16 1
A039756 \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 -1 1 -4 1 1 -9 13 -1 1 -16 58 -40 1
A049308 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 4 40 640 14080 394240 13404160
A109692 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 1 1 4 3 1 9 23 15 1 16 86 176 105 1
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -1 1 -4 3 1 -9 23 -15 1 -16 86 -176
A197130 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 1 10 100 1136 14816 220032 3679488
A202153 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 2 5 13 40 127 443 1610 6207 24919
A203159 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 6 44 400 4384 56448 836352 14026752
A293318 \(T_{2 n, n}\) ➤ CentralE ➤ 1 4 86 3480 208054 16486680 1628301884
A370912 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 15 48 105 192 315 480 693 960 1287 1680
A374866 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 2 15 192 3465 80640 2297295 77414400