Binomialbell
A056857 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k} \text{Bell}(n-k) } \)
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
A000027 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
---------- \(\sum_{k=0}^{1} T_{1, k}\ n^k\) ➤ 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
A000110 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 1 2 5 15 52 203 877 4140 21147 115975
---------- \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 2 5 15 52 203 877 4140 21147 115975
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 2 5 15 52 203 877 4140 21147 115975
A000296 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 0 1 1 4 11 41 162 715 3425 17722
A001844 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 5 13 25 41 61 85 113 145 181 221 265
A002378 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 2 6 12 20 30 42 56 72 90 110 132 156
A002522 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 2 5 10 17 26 37 50 65 82 101 122 145
A005491 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 5 15 37 77 141 235 365 537 757 1031
A005493 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 3 10 37 151 674 3263 17007 94828
A005494 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 4 17 77 372 1915 10481 60814 372939
A052889 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 2 6 20 75 312 1421 7016 37260 211470
A056857 \(T_{n, k}\) ➤ Triangle ➤ 1 1 1 2 2 1 5 6 3 1 15 20 12 4 1 52 75
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 1 -1 2 -2 1 5 -6 3 -1 15 -20 12 -4 1
A056860 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 1 1 2 2 1 3 6 5 1 4 12 20 15 1 5 20
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -1 1 -2 2 1 -3 6 -5 1 -4 12 -20 15
A070071 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 4 15 60 260 1218 6139 33120 190323
A102286 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 2 7 24 96 418 1989 10216 56275
A105479 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 3 12 50 225 1092 5684 31572 186300
A105480 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 4 20 100 525 2912 17052 105240 683100
A124102 \(T_{2 n, n}\) ➤ CentralE ➤ 1 2 12 100 1050 13104 187572 3009864
A124311 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -1 5 -21 121 -793 5917 -49101 447153
A124427 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 3 9 30 112 463 2095 10279 54267
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 3 9 30 112 463 2095 10279 54267
A126390 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 3 13 71 457 3355 27509 248127 2434129
A126617 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -1 2 -3 7 -10 31 -21 204 307 2811
A127741 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 1 6 30 148 755 4044 22841 136056
A129334 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 -1 1 0 -2 1 1 0 -3 1 1 4 0 -4 1 -2 5
A134481 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 5 20 50 100 175 280 420 600 825 1100
A134980 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 2 10 77 799 10427 163967 3017562
A143987 \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 -1 1 -2 0 1 -3 0 1 1 -4 0 4 1 1 -5
A175716 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 6 27 120 560 2778 14665 82232
A175757 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 2 1 6 3 1 20 12 4 1 75 50 20 5 1 312
A224271 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 1 3 8 28 107 459 2151 10931 59700
A284859 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 4 25 199 1876 20257 245017 3266914
A297926 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 1 6 50 525 6552 93786 1504932 26640900
A303586 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 2 3 6 11 23 47 103 226 518 1200
A307066 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 2 13 199 5329 216151 12211597