Binarypell
A038207 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k} \, 2^{n-k} } \)
A000007 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 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
---------- \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 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 ➤ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
A000079 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 2 4 8 16 32 64 128 256 512 1024 2048
A000129 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 2 5 12 29 70 169 408 985 2378 5741
A000244 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 3 9 27 81 243 729 2187 6561 19683
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 3 9 27 81 243 729 2187 6561 19683
---------- \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -3 9 -27 81 -243 729 -2187 6561
A000272 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 3 16 125 1296 16807 262144 4782969
A000290 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 4 9 16 25 36 49 64 81 100 121 144 169
A000302 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 4 16 64 256 1024 4096 16384 65536
A000351 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 5 25 125 625 3125 15625 78125 390625
---------- \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 5 25 125 625 3125 15625 78125 390625
A000420 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 7 49 343 2401 16807 117649 823543
A000578 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 8 27 64 125 216 343 512 729 1000 1331
A001045 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 3 5 11 21 43 85 171 341 683 1365
A001787 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 4 12 32 80 192 448 1024 2304 5120
A001788 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 6 24 80 240 672 1792 4608 11520 28160
A001789 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 8 40 160 560 1792 5376 15360 42240
A001850 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 3 13 63 321 1683 8989 48639 265729
A003462 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 4 13 40 121 364 1093 3280 9841
A005408 \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ RevPolyRow1 ➤ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
A005843 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 2 4 6 8 10 12 14 16 18 20 22 24 26 28
A006234 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 4 15 54 189 648 2187 7290 24057 78732
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 4 15 54 189 648 2187 7290 24057 78732
A007051 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 2 5 14 41 122 365 1094 3281 9842
A013609 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 2 1 4 4 1 6 12 8 1 8 24 32 16 1 10
---------- \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 -2 1 -4 4 1 -6 12 -8 1 -8 24 -32 16
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -2 1 -4 4 1 -6 12 -8 1 -8 24 -32 16
A016754 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 9 25 49 81 121 169 225 289 361 441
A016755 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 27 125 343 729 1331 2197 3375 4913
A027471 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 6 27 108 405 1458 5103 17496 59049
A038207 \(T_{n, k}\) ➤ Triangle ➤ 1 2 1 4 4 1 8 12 6 1 16 32 24 8 1 32 80
---------- \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 -2 1 4 -4 1 -8 12 -6 1 16 -32 24 -8 1
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 2 -1 4 -4 1 8 -12 6 -1 16 -32 24 -8 1
A046092 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 4 12 24 40 60 84 112 144 180 220 264
A046717 \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 5 13 41 121 365 1093 3281 9841
A059304 \(T_{2 n, n}\) ➤ CentralE ➤ 1 4 24 160 1120 8064 59136 439296
A062189 \(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 2 20 126 648 2970 12636 51030 198288
A069720 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 6 40 280 2016 14784 109824 823680
A069723 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 2 12 80 560 4032 29568 219648 1647360
A081038 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 5 21 81 297 1053 3645 12393 41553
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 5 21 81 297 1053 3645 12393 41553
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 5 21 81 297 1053 3645 12393 41553
A085527 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 3 25 343 6561 161051 4826809
A098660 \(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 4 6 24 40 160 280 1120 2016 8064
A109388 \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 2 4 12 32 80 240 672 1792 5376 15360
A128099 \(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 2 1 4 1 6 4 1 8 12 1 10 24 8 1 12
A130809 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 8 32 80 160 280 448 672 960 1320 1760
A152011 \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 2 4 14 40 122 364 1094 3280 9842
A171977 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 2 4 2 8 2 4 2 16 2 4 2 8 2 4 2 32 2 4
A207538 \(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 2 4 1 8 4 16 12 1 32 32 6 64 80 24 1
A212697 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 2 12 54 216 810 2916 10206 34992
A276985 \(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 2 4 4 6 12 8 8 24 32 16 10 40 80 80 32