Doublepochhammer
A039683 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ [x^k]\, x(x-2)(x-4)...(x-2n+2) } \)
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
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 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
A000165 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 2 8 48 384 3840 46080 645120 10321920
---------- \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 8 48 384 3840 46080 645120 10321920
A001147 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 3 15 105 945 10395 135135 2027025
---------- \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 -1 -3 -15 -105 -945 -10395 -135135
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 3 15 105 945 10395 135135 2027025
---------- \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 15 105 945 10395 135135 2027025
A001193 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 2 9 60 525 5670 72765 1081080
A001879 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 1 6 45 420 4725 62370 945945
A002378 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 2 6 12 20 30 42 56 72 90 110 132 156
A004041 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 4 23 176 1689 19524 264207 4098240
A005408 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
A005563 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 3 8 15 24 35 48 63 80 99 120 143 168
A007696 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 5 45 585 9945 208845 5221125
A008542 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 7 91 1729 43225 1339975 49579075
A008545 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 -3 21 -231 3465 -65835 1514205
A014634 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 15 45 91 153 231 325 435 561 703 861
A036464 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 8 44 140 340 700 1288 2184 3480 5280
A039683 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 2 1 8 6 1 48 44 12 1 384 400 140 20 1
A065176 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 2 2 4 4 2 2 8 8 2 2 4 4 2 2 16 16 2
A075497 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -2 1 4 -6 1 -8 28 -12 1 16 -120 100
A113551 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 8 105 1920 45045 1290240 43648605
A114160 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 7 38 281 2634 29919 399342 6125265
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 7 38 281 2634 29919 399342 6125265
A203159 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 6 44 400 4384 56448 836352 14026752
A353255 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 2 9 54 429 4252 50605 703388
A370912 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 15 48 105 192 315 480 693 960 1287