Delannoy
A008288 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \text{Hyper}([-k, k - n], [1], 2) } \)
A000012 \(T_{n , 0}\) ➤ TablCol0 ➤ 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
A000027 \(\sum_{k=0}^{1} T_{1, k}\ n^k\) ➤ 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
A000035 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 0 -1 0 1 0 -1 0 1 0 -1 0 1 0 -1 0 1 0
A000073 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 1 2 4 7 13 24 44 81 149 274 504 927
---------- \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 2 4 7 13 24 44 81 149 274 504 927
A000129 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 2 5 12 29 70 169 408 985 2378 5741
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 2 5 12 29 70 169 408 985 2378 5741
A001844 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 5 13 25 41 61 85 113 145 181 221 265
---------- \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 1 5 13 25 41 61 85 113 145 181 221 265
A001845 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 7 25 63 129 231 377 575 833 1159 1561
---------- \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 1 7 25 63 129 231 377 575 833 1159 1561
A001850 \(T_{2 n, n}\) ➤ CentralE ➤ 1 3 13 63 321 1683 8989 48639 265729
A002002 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 1 5 25 129 681 3653 19825 108545 598417
---------- \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 5 25 129 681 3653 19825 108545 598417
A005408 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
---------- \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
A006139 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 2 8 32 136 592 2624 11776 53344
A007482 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 3 11 39 139 495 1763 6279 22363 79647
---------- \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 3 11 39 139 495 1763 6279 22363 79647
A008288 \(T_{n, k}\) ➤ Triangle ➤ 1 1 1 1 3 1 1 5 5 1 1 7 13 7 1 1 9 25
---------- \(T_{n, n - k}\) ➤ Trev ➤ 1 1 1 1 3 1 1 5 5 1 1 7 13 7 1 1 9 25
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 1 -1 1 -3 1 1 -5 5 -1 1 -7 13 -7 1 1
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -1 1 -3 1 1 -5 5 -1 1 -7 13 -7 1 1
A015530 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 4 19 88 409 1900 8827 41008 190513
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 4 19 88 409 1900 8827 41008 190513
A026003 \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 3 5 13 25 63 129 321 681 1683 3653
---------- \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 1 3 5 13 25 63 129 321 681 1683 3653
---------- \(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 3 5 13 25 63 129 321 681 1683 3653
A026937 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 3 10 30 87 245 676 1836 4925 13079
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 3 10 30 87 245 676 1836 4925 13079
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 3 10 30 87 245 676 1836 4925 13079
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 3 10 30 87 245 676 1836 4925 13079
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 3 10 30 87 245 676 1836 4925 13079
A028387 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 1 5 11 19 29 41 55 71 89 109 131 155
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 5 11 19 29 41 55 71 89 109 131 155
A033878 \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 -1 1 -3 2 1 -5 10 -6 1 -7 22 -38 22
A059304 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 0 -4 0 24 0 -160 0 1120 0 -8064 0
A077020 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -1 -1 3 -1 -5 7 3 -17 11 23 -45 -1 91
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -1 -1 3 -1 -5 7 3 -17 11 23 -45 -1 91
A116404 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 1 2 6 15 35 84 204 493 1189 2870 6930
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 2 6 15 35 84 204 493 1189 2870 6930
A132372 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 -1 1 2 -3 1 -6 10 -5 1 22 -38 22 -7 1
---------- \((T_{n, n - k})^{-1}\) ➤ Tinvrev ➤ 1 -1 1 2 -3 1 -6 10 -5 1 22 -38 22 -7 1
A364553 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 5 18 58 175 507 1428 3940 10701
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 1 5 18 58 175 507 1428 3940 10701
A372443 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 3 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A376871 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 2 11 88 941 12546 200479 3735264
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 2 11 88 941 12546 200479 3735264