Schroeder
A122538 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ is(k = 0)\ ? \ 0^{n} : T(n-1,k-1)+T(n-1,k)+T(n,k+1) } \)
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 \(T_{n , n}\) ➤ 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
A000225 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 1 -3 7 -15 31 -63 127 -255 511 -1023
A001003 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 3 11 45 197 903 4279 20793 103049
---------- \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 -1 -3 -11 -45 -197 -903 -4279
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 3 11 45 197 903 4279 20793 103049
A005408 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
A005563 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 3 8 15 24 35 48 63 80 99 120 143 168
A005843 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
A006318 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 2 6 22 90 394 1806 8558 41586 206098
---------- \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 2 6 22 90 394 1806 8558 41586
A006319 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 4 16 68 304 1412 6752 33028 164512
A006603 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 2 7 26 107 468 2141 10124 49101
A010683 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 2 7 28 121 550 2591 12536 61921
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 7 28 121 550 2591 12536 61921
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 7 28 121 550 2591 12536 61921
A026003 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 3 5 13 25 63 129 321 681 1683
A033877 \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 2 1 4 6 1 6 16 22 1 8 30 68 90 1 10
A035607 \(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -2 1 -4 2 1 -6 8 -2 1 -8 18 -12 2 1
A054000 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 6 16 30 48 70 96 126 160 198 240 286
A055642 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
A065096 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 6 31 156 785 3978 20335 104856
A080247 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 2 1 6 4 1 22 16 6 1 90 68 30 8 1 394
A080859 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 11 33 67 113 171 241 323 417 523 641
A103885 \(T_{2 n, n}\) ➤ CentralE ➤ 1 2 16 146 1408 14002 142000 1459810
A106579 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 0 1 2 0 1 4 6 0 1 6 16 22 0 1 8 30
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 0 1 -2 0 1 -4 6 0 1 -6 16 -22 0 1
A109980 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 8 36 172 852 4324 22332 116876
A113413 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -2 1 2 -4 1 -2 8 -6 1 2 -12 18 -8 1
A122538 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 2 1 0 6 4 1 0 22 16 6 1 0 90 68
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 -2 1 0 -6 4 -1 0 -22 16 -6 1 0
A122542 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 0 -2 1 0 2 -4 1 0 -2 8 -6 1 0 2
A144944 \(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 1 1 3 3 1 5 11 11 1 7 23 45 45 1 9
A178792 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 5 31 209 1471 10625 78079 580865
A227506 \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 1 7 17 121 353 2591 8257 61921
A239204 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 1 4 17 76 353 1688 8257 41128
---------- \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 4 17 76 353 1688 8257 41128 207905
A266213 \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 -2 0 1 -4 2 0 1 -6 8 -2 0 1 -8
A330801 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 4 30 264 2490 24396 244790 2496528
A330802 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 5 33 253 2121 18853 174609 1667021
A330803 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 -3 17 -123 1001 -8739 79969 -756939