Stirlingset
A048993 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ {n \brace k} } \)
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 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n , n}\) ➤ 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
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A000110 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 2 5 15 52 203 877 4140 21147 115975
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 2 5 15 52 203 877 4140 21147 115975
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 5 15 52 203 877 4140 21147 115975
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 5 15 52 203 877 4140 21147 115975
A000217 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 1 3 6 10 15 21 28 36 45 55 66 78 91
A000225 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 3 7 15 31 63 127 255 511 1023 2047
A000392 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 6 25 90 301 966 3025 9330 28501 86526
A000587 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 0 1 1 -2 -9 -9 50 267 413 -2180
A001296 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 1 7 25 65 140 266 462 750 1155 1705
A001297 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 1 15 90 350 1050 2646 5880 11880
A001861 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 6 22 94 454 2430 14214 89918 610182
A002378 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 2 6 12 20 30 42 56 72 90 110 132 156
A002870 \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 1 3 7 25 90 350 1701 7770 42525
A004211 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 3 11 49 257 1539 10299 75905 609441
A004212 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 4 19 109 742 5815 51193 498118
A005493 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 3 10 37 151 674 3263 17007 94828
A007820 \(T_{2 n, n}\) ➤ CentralE ➤ 1 1 7 90 1701 42525 1323652 49329280
A008277 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 1 1 1 3 1 1 7 6 1 1 15 25 10 1 1 31
A008278 \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 1 1 3 1 1 6 7 1 1 10 25 15 1 1 15
A009235 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 -1 -1 9 -23 -25 583 -3087 4401
A024428 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 2 4 8 18 42 102 260 684 1860 5216
A024429 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 1 2 7 27 106 443 2045 10440 57781
A024430 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 1 3 8 25 97 434 2095 10707 58194
A027710 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 12 57 309 1866 12351 88563 681870
A028387 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 5 11 19 29 41 55 71 89 109 131 155
A033445 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 5 22 57 116 205 330 497 712 981 1310
A033452 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 5 22 99 471 2386 12867 73681 446620
A048993 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 1 1 0 1 3 1 0 1 7 6 1 0 1 15 25
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 -1 1 0 -1 3 -1 0 -1 7 -6 1 0
A054654 \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 -1 0 1 -3 2 0 1 -6 11 -6 0 1
A063040 \(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 1 3 42 150 36270 270900 9440379900
A089026 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 1 3 1 5 1 7 1 1 1 11 1 13 1 1 1 17
A094638 \(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -1 1 -3 2 1 -6 11 -6 1 -10 35 -50
A096647 \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 1 2 8 27 97 443 2095 10440 58194
A096648 \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 1 3 7 25 106 434 2045 10707 57781
A106342 \((T_{n + 1, n - k + 1})^{-1}\) ➤ Tinvrev11 ➤ 1 -1 1 2 -3 1 -9 15 -7 1 94 -160 80 -15
A106800 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 0 1 1 0 1 3 1 0 1 6 7 1 0 1 10 25
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 0 1 -1 0 1 -3 1 0 1 -6 7 -1 0 1 -10
A122455 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 3 13 71 456 3337 27203 243203
A129506 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 3 25 350 6951 179487 5715424
A130534 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -1 1 2 -3 1 -6 11 -6 1 24 -50 35 -10
A132393 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 0 -1 1 0 2 -3 1 0 -6 11 -6 1 0 24
A171367 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 1 2 4 9 22 58 164 495 1587 5379
A213170 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 2 2 -6 -14 26 178 90 -2382 -9446
A242817 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 6 57 756 12880 268098 6593839
A247238 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 1 15 301 7770 246730 9321312
A278677 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 1 5 23 109 544 2876 16113 95495
A301419 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 3 19 201 3176 69823 2026249
A321331 \(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 1 2 1 6 3 1 14 18 4 1 30 75 40 5 1 62
A343278 \(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 1 3 7 25 90 350 1701 6951 42525
A343279 \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 1 1 7 15 90 301 1701 7770 42525
A343841 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 1 -1 -5 15 56 -455 -237 16947 -64220
A359107 \(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 0 1 0 1 2 0 1 4 5 0 1 8 14 15 0 1 16
A359109 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 3 10 38 161 747 3753 20253 116642
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 3 10 38 161 747 3753 20253 116642
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 3 10 38 161 747 3753 20253 116642