Monotone
A059481 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n+k-1}{k} } \)
A000007 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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
---------- \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A000027 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 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
A000217 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 3 6 10 15 21 28 36 45 55 66 78 91 105
A000292 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 10 20 35 56 84 120 165 220 286 364 455
A000984 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 1 2 6 20 70 252 924 3432 12870 48620
---------- \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 2 6 20 70 252 924 3432 12870 48620
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 2 6 20 70 252 924 3432 12870 48620
A001700 \(T_{n , n}\) ➤ TablDiag0 ➤ 1 1 3 10 35 126 462 1716 6435 24310
---------- \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 1 3 10 35 126 462 1716 6435 24310 92378
---------- \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 3 10 35 126 462 1716 6435 24310
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 3 10 35 126 462 1716 6435 24310 92378
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 3 10 35 126 462 1716 6435 24310 92378
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 3 10 35 126 462 1716 6435 24310 92378
A001791 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 1 4 15 56 210 792 3003 11440 43758
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 1 4 15 56 210 792 3003 11440 43758
A002003 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 2 8 38 192 1002 5336 28814 157184
A005809 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 1 3 15 84 495 3003 18564 116280 735471
A014300 \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 1 2 7 24 86 314 1163 4352 16414 62292
A026641 \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 4 13 46 166 610 2269 8518 32206
A032443 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 3 11 42 163 638 2510 9908 39203
A034275 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 3 14 65 294 1302 5676 24453 104390
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 3 14 65 294 1302 5676 24453 104390
A046899 \(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 1 2 1 3 6 1 4 10 20 1 5 15 35 70 1 6
---------- \(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 1 1 2 1 3 6 1 4 10 20 1 5 15 35 70 1 6
A056109 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 1 6 17 34 57 86 121 162 209 262 321 386
A059100 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 3 6 11 18 27 38 51 66 83 102 123 146
A059481 \(T_{n, k}\) ➤ Triangle ➤ 1 1 1 1 2 3 1 3 6 10 1 4 10 20 35 1 5
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 1 -1 1 -2 3 1 -3 6 -10 1 -4 10 -20 35
A072547 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 0 2 -6 22 -80 296 -1106 4166 -15792
A081204 \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 1 2 3 10 15 56 84 330 495 2002 3003
A092392 \(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 1 2 1 6 3 1 20 10 4 1 70 35 15 5 1 252
A100100 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 1 3 2 1 10 6 3 1 35 20 10 4 1 126
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -1 3 -2 1 10 -6 3 -1 35 -20 10 -4 1
A100192 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 4 18 82 374 1704 7752 35214 159750
A100217 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 4 12 42 149 543 2007 7501 28265
A100218 \((T_{n, n - k})^{-1}\) ➤ Tinvrev ➤ 1 -1 1 -1 -2 1 -1 0 -3 1 -1 0 2 -4 1 -1
A110609 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 8 45 224 1050 4752 21021 91520
A113214 \(T^{-1}_{n + 1, n-k}\) ➤ RevTinv11 ➤ 1 -2 1 0 -3 1 0 2 -4 1 0 0 5 -5 1 0 0
A116406 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 1 2 3 7 11 26 42 99 163 382 638 1486
A119259 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 3 17 111 769 5503 40193 297727
A165257 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 2 3 3 6 10 4 10 20 35 5 15 35 70 126
A165817 \(T_{2 n, n}\) ➤ CentralE ➤ 1 2 10 56 330 2002 12376 77520 490314
A220101 \(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 1 6 27 112 450 1782 7007 27456 107406
A293574 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 2 11 82 787 9476 139134 2422218
A355341 \(T^{-1}_{n + 1, n-k}\) ➤ RevTrevinv11 ➤ 1 1 -2 1 -3 0 1 -4 2 0 1 -5 5 0 0 1 -6
A367548 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -1 3 2 19 54 222 804 3075 11630 44458
A368488 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 2 17 334 10417 442276 23690809