Motzkin
A064189 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k} \text{Hyper}([(k-n)/2, (k-n+1)/2], [k+2], 4) } \)
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
---------- \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 2 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
A000096 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 2 5 9 14 20 27 35 44 54 65 77 90 104
A000244 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 3 9 27 81 243 729 2187 6561 19683
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 3 9 27 81 243 729 2187 6561 19683
A000297 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 4 12 25 44 70 104 147 200 264 340 429
A001006 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 1 2 4 9 21 51 127 323 835 2188 5798
A001844 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 5 13 25 41 61 85 113 145 181 221 265
A002026 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 2 5 12 30 76 196 512 1353 3610 9713
A002426 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 1 3 7 19 51 141 393 1107 3139 8953
A002522 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 2 5 10 17 26 37 50 65 82 101 122 145
A005043 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 0 1 1 3 6 15 36 91 232 603 1585 4213
---------- \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 1 3 6 15 36 91 232 603 1585 4213
A005322 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 3 9 25 69 189 518 1422 3915 10813
A005323 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 4 14 44 133 392 1140 3288 9438 27016
A005717 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 2 6 16 45 126 357 1016 2907 8350
A005773 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 2 5 13 35 96 267 750 2123 6046 17303
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 2 5 13 35 96 267 750 2123 6046 17303
A026300 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 1 1 2 2 1 3 5 4 1 4 9 12 9 1 5 14
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -1 1 -2 2 1 -3 5 -4 1 -4 9 -12 9 1
A026302 \(T_{2 n, n}\) ➤ CentralE ➤ 1 2 9 44 230 1242 6853 38376 217242
A026307 \(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 2 3 9 14 44 70 230 369 1242 2002
A026938 \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 2 5 12 30 76 196 518 1422 3915
A026943 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 3 11 38 129 429 1407 4563 14669 46823
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 3 11 38 129 429 1407 4563 14669 46823
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 3 11 38 129 429 1407 4563 14669 46823
A059738 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 3 10 34 117 405 1407 4899 17083 59629
A064189 \(T_{n, k}\) ➤ Triangle ➤ 1 1 1 2 2 1 4 5 3 1 9 12 9 4 1 21 30 25
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 1 -1 2 -2 1 4 -5 3 -1 9 -12 9 -4 1 21
A101950 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 -1 1 0 -2 1 1 1 -3 1 -1 2 3 -4 1 0 -4
A106489 \(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 1 2 1 4 2 9 5 1 21 12 3 51 30 9 1 127
A135859 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 4 13 34 73 136 229 358 529 748 1021
A191519 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 2 3 6 10 19 33 62 110 205 368 683
A327871 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 3 14 70 369 2002 11076 62127 352070
A330796 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 4 14 46 147 462 1437 4438 13637
A330799 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 3 13 59 285 1419 7245 37659 198589
A330800 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -1 5 -17 77 -345 1653 -8097 40733
A344394 \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 1 2 5 9 25 44 133 230 726 1242 4037
A344396 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 1 5 25 133 726 4037 22737 129285 740554
A344502 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 2 7 29 128 587 2759 13190 63844
A344503 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 0 -1 3 0 -5 15 0 -28 84 0 -165 495 0
A344504 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 6 26 100 361 1254 4245 14108 46247
A344506 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 4 17 73 315 1362 5895 25528 110579
A344507 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -1 2 -2 5 -3 15 3 59 73 308 632 1951