Bessel
A132062 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ 2^{k - n} \binom{2n - 2k}{n - k} \binom{2n - k - 1}{k - 1} (n - k)! } \)
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 0 0 1 0 1 0 3 1 0 15 3 0 105 15 1 0
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 0 1 0 1 2 0 3 6 7 0 15 30 36 37 0 105
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 1 2 3 6 3 15 30 18 4 105 210 135 40 5
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 1 3 30 630 3780 207900 8108100
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 3 16 118 1101 12421 164473 2501661
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 5 19 104 761 7027 78590 1033559
\(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 1 4 18 121 1056 11386 145960
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 1 3 15 105 420 4725 17325 270270
\(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 3 105 4725 270270 18918900 1571349780
\(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 5 19 104 761 7027 78590 1033559
\(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 5 24 145 1115 10596 120715 1607999
\(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 3 19 175 2076 29911 505093 9757539
\(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 1 -1 1 7 -124 1591 -19991 256691
\(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 6 63 940 18150 429786 12051697
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 0 1 0 3 3 0 6 15 15 0 10 45 105 105 0
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 0 0 1 0 3 3 0 15 15 6 0 105 105 45 10 0
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 0 1 1 1 3 0 1 6 3 1 10 15 0 1 15
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 1 1 2 2 1 4 7 7 1 7 22 37 37 1 11
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 0 1 0 3 6 0 6 30 45 0 10 90 315 420 0
\(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 1 4 16 151 1051 15331 152839
\(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 1 3 21 115 1380 11676 200683
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 3 16 118 1101 12421 164473 2501661
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 1 3 15 45 420 1260 17325 51975
\(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 2 -2 -2 -22 -206 -2354 -31426
\(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 3 16 118 1101 12421 164473 2501661
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 0 1 15 201 2815 42960 721896 13352815
\(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 3 37 1225 79926 8667631 1409457463