Stirlingcycle
A132393 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ {n \brack k} } \)
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
---------- \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 2 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
---------- \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ 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 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 1 2 6 24 120 720 5040 40320 362880
---------- \(\sum_{k=0}^{n} T_{n, k}\) ➤ 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
---------- \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 6 24 120 720 5040 40320 362880
A000217 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 1 3 6 10 15 21 28 36 45 55 66 78 91
A000254 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 3 11 50 274 1764 13068 109584 1026576
---------- \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 3 11 50 274 1764 13068 109584
A000384 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 6 15 28 45 66 91 120 153 190 231 276
A000399 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 6 35 225 1624 13132 118124 1172700
A000407 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 6 60 840 15120 332640 8648640
A000774 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 5 17 74 394 2484 18108 149904
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 5 17 74 394 2484 18108 149904
A000914 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 2 11 35 85 175 322 546 870 1320 1925
A001147 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 3 15 105 945 10395 135135 2027025
---------- \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 -1 3 -15 105 -945 10395 -135135
A001303 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 6 50 225 735 1960 4536 9450 18150
A001710 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 1 3 12 60 360 2520 20160 181440
---------- \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 1 3 12 60 360 2520 20160 181440
---------- \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 12 60 360 2520 20160 181440 1814400
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 1 3 12 60 360 2520 20160 181440
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 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
A007531 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 6 24 60 120 210 336 504 720 990 1320
A007559 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 4 28 280 3640 58240 1106560
A008277 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -1 1 1 -3 1 -1 7 -6 1 1 -15 25 -10 1
A008278 \(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -1 1 -3 1 1 -6 7 -1 1 -10 25 -15 1
A028421 \(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 1 2 2 6 3 6 22 18 4 24 100 105 40 5
A048993 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 0 -1 1 0 1 -3 1 0 -1 7 -6 1 0 1
A054654 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 0 1 1 0 1 3 2 0 1 6 11 6 0 1 10 35
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 0 1 -1 0 1 -3 2 0 1 -6 11 -6 0 1
A063039 \(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 1 6 66 4200 4192200 5115600
A065048 \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 1 3 11 50 274 1764 13132 118124
A067318 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 1 7 46 326 2556 22212 212976
A092985 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 3 28 585 22176 1339975 118514880
A094638 \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 1 1 3 2 1 6 11 6 1 10 35 50 24 1 15
A096747 \(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 1 1 2 2 1 4 6 6 1 7 18 24 24 1 11
A106800 \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 -1 0 1 -3 1 0 1 -6 7 -1 0 1 -10
A121586 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 3 13 70 446 3276 27252 253296
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 3 13 70 446 3276 27252 253296
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 3 13 70 446 3276 27252 253296
A124380 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 2 4 9 22 57 157 453 1368 4296
A129505 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 3 35 735 22449 902055 44990231
A130534 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 1 1 2 3 1 6 11 6 1 24 50 35 10 1 120
A132393 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 1 1 0 2 3 1 0 6 11 6 1 0 24 50
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 -1 1 0 -2 3 -1 0 -6 11 -6 1 0
A151881 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 5 23 120 724 5012 39332 345832
A154415 \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 1 2 11 50 225 1624 6769 67284
A187646 \(T_{2 n, n}\) ➤ CentralE ➤ 1 1 11 225 6769 269325 13339535
A211210 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 3 16 115 1021 10696 128472 1734447
A317274 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 1 -1 -2 19 -79 76 2640 -36945 329371
A331327 \(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 0 0 1 0 1 0 2 1 0 6 3 0 24 11 1 0 120
A343579 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 1 3 9 36 176 1030 7039 55098
A349782 \(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 0 1 0 1 2 0 2 5 6 0 6 17 23 24 0 24
A367777 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 2 50 1624 67284 3416930 206070150