Charlier
A046716 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \sum_{j=0}^{k} (-1)^k \, \binom{n}{k-j}\,{j+n-k \brack n-k} } \)
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
---------- \(T_{n , n}\) ➤ TablDiag0 ➤ 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
A000027 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 0 -1 2 -3 4 -5 6 -7 8 -9 10 -11 12
---------- \(\sum_{k=0}^{1} T_{1, k}\ n^k\) ➤ PolyRow1 ➤ 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11
---------- \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ RevPolyRow1 ➤ -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
A000217 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ -1 -3 -6 -10 -15 -21 -28 -36 -45 -55
A000522 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 2 5 16 65 326 1957 13700 109601
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 2 5 16 65 326 1957 13700 109601
A001339 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -3 11 -49 261 -1631 11743 -95901
A002104 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 1 -3 8 -24 89 -415 2372 -16072 125673
A009132 \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 -1 2 -7 31 -161 976 -6847 54797
A009578 \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 1 -3 9 -34 165 -981 6853 -54804
A009940 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 0 -1 -4 -15 -56 -185 -204 6209 112400
A028387 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 -1 -1 5 -11 19 -29 41 -55 71 -89
---------- \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 1 -1 -1 1 5 11 19 29 41 55 71 89 109
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 -1 -1 1 5 11 19 29 41 55 71 89 109
A046716 \(T_{n, k}\) ➤ Triangle ➤ 1 1 -1 1 -3 1 1 -6 8 -1 1 -10 29 -24 1
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 1 1 1 3 1 1 6 8 1 1 10 29 24 1 1 15
A049020 \((T_{n, n - k})^{-1}\) ➤ Tinvrev ➤ 1 1 1 2 3 1 5 10 6 1 15 37 31 10 1 52
A081367 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 3 11 53 345 2947 31411 400437 5927921
A094816 \(T_{n, n - k}\) ➤ Trev ➤ 1 -1 1 1 -3 1 -1 8 -6 1 1 -24 29 -10 1
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 -1 -1 1 3 1 -1 -8 -6 -1 1 24 29 10 1
A174965 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A290312 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 8 29 75 160 301 518 834 1275 1870
A290313 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ -1 -24 -145 -545 -1575 -3836 -8274