Bessel
A132062 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ 2^{k - n} \binom{2n - 2k}{n - k} \binom{2n - k - 1}{k - 1} (n - k)! } \)
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
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A000085 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 2 4 10 26 76 232 764 2620 9496
A000217 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 1 3 6 10 15 21 28 36 45 55 66 78 91
A001147 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 1 3 15 105 945 10395 135135 2027025
---------- \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 3 15 105 945 10395 135135 2027025
---------- \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 1 3 15 105 945 10395 135135 2027025
A001497 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 1 1 3 3 1 15 15 6 1 105 105 45 10 1
A001498 \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 1 1 3 3 1 6 15 15 1 10 45 105 105 1
A001514 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 1 9 81 835 9990 137466 2148139
A001515 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 2 7 37 266 2431 27007 353522
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 2 7 37 266 2431 27007 353522
A001517 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 3 19 193 2721 49171 1084483
A001518 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 4 37 559 11776 318511 10522639
A001879 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 6 45 420 4725 62370 945945 16216200
A002119 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 -1 7 -71 1001 -18089 398959
A002378 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 2 6 12 20 30 42 56 72 90 110 132 156
A003215 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 7 19 37 61 91 127 169 217 271 331 397
A025164 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 1 4 21 151 1380 15331 200683
A036244 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 1 3 16 115 1051 11676 152839
A049403 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -1 1 0 -3 1 0 3 -6 1 0 0 15 -10 1 0 0
A050534 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 3 15 45 105 210 378 630 990 1485 2145
A068601 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 7 26 63 124 215 342 511 728 999 1330
A069834 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 1 3 3 5 15 21 7 9 45 55 33 39 91
A104548 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 0 1 1 0 1 3 3 0 1 6 15 15 0 1 10 45
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 0 1 -1 0 1 -3 3 0 1 -6 15 -15 0 1
A107103 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 3 12 67 495 4596 51583 680037
A107104 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 6 26 154 1182 11254 128522 1715802
A111924 \(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -1 1 -3 0 1 -6 3 0 1 -10 15 0 0 1
A122848 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 0 -1 1 0 0 -3 1 0 0 3 -6 1 0 0 0
A132062 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 1 1 0 3 3 1 0 15 15 6 1 0 105
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 -1 1 0 -3 3 -1 0 -15 15 -6 1 0
A144299 \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 -1 0 1 -3 0 0 1 -6 3 0 0 1 -10
A240440 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 15 105 420 1260 3150 6930 13860 25740
A245066 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 3 45 1260 51975 2837835 192972780
A278990 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 0 -1 -5 -36 -329 -3655 -47844
A369746 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 12 63 423 3528 35559 422901 5817744
A376872 \(T_{2 n, n}\) ➤ CentralE ➤ 1 1 15 420 17325 945945 64324260