Doublepochhammer
A039683 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ [x^k]\, x(x-2)(x-4)...(x-2n+2) } \)
\(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 2 1 0 8 6 1 0 48 44 12 1 0 384
\(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 0 -2 1 0 4 -6 1 0 -8 28 -12 1 0
\(T_{n, n - k}\) ➤ Trev ➤ 1 1 0 1 2 0 1 6 8 0 1 12 44 48 0 1 20
\(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 -2 0 1 -6 4 0 1 -12 28 -8 0 1
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 2 1 6 8 1 12 44 48 1 20 140 400 384
\(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 -2 1 -6 4 1 -12 28 -8 1 -20 100
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 0 0 1 0 2 0 8 1 0 48 6 0 384 44 1 0
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 0 1 0 2 3 0 8 14 15 0 48 92 104 105 0
\(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 -2 1 0 -8 6 -1 0 -48 44 -12 1
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 2 2 8 12 3 48 88 36 4 384 800 420 80
\(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 12 140 1800 25984 420224 7559936
\(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 48 400 1800 5880 15680 36288 75600
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 2 24 528 67200 134150400 327398400
\(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 2 8 48 400 4384 56448 836352
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 5 37 349 3981 53241 816873 14144985
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 2 8 44 400 1800 25984 108304
\(T_{2 n, n}\) ➤ CentralE ➤ 1 2 44 1800 108304 8618400 853730240
\(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 8 400 25984 2153088 218683520
\(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 6 41 348 3589 43802 618869 9946584
\(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 5 43 505 7421 130081 2638203
\(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 1 -3 7 25 -779 11641 -144801 1463169
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 0 2 0 6 8 0 12 44 48 0 20 140 400 384 0
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 0 0 2 0 8 6 0 48 44 12 0 384 400 140 20
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 0 1 2 1 6 0 1 12 8 1 20 44 0 1 30
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 1 1 3 3 1 7 15 15 1 13 57 105 105 1
\(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 0 1 -2 0 1 -6 8 0 1 -12 44 -48 0 1
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 0 2 0 6 16 0 12 88 144 0 20 280 1200
\(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 1 9 45 525 4725 72765 945945
\(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 2 6 60 420 5670 62370 1081080
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 5 37 349 3981 53241 816873 14144985
\(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 3 7 21 65 219 783 2941 11625
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 2 6 44 140 1800 5880 108304 359184
\(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 6 140 5880 359184 28865760 2879374784
\(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 2 22 244 3036 42846 681738 12117960
\(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 5 37 349 3981 53241 816873 14144985
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 0 2 38 620 10324 183734 3541586
\(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 5 91 3825 293601 35942725