Charlier
A046716 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \sum_{j=0}^{k} (-1)^k \, \binom{n}{k-j}\,{j+n-k \brack n-k} } \)
\(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 -1 1 -4 3 1 25 -18 -8 1 705 -509 -221
\(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 -1 1 3 -4 1 -8 -18 25 1 24 -221
\(T_{n + 1, k + 1} \) ➤ Toff11 ➤ -1 -3 1 -6 8 -1 -10 29 -24 1 -15 75
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ -1 1 -3 -1 8 -6 1 -24 29 -10 -1 89 -145
\(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 3 1 -18 -8 1 -509 -221 24 1 42481
\(T^{-1}_{n + 1, n - k + 1}\) ➤ Trevinv11 ➤ 1 1 3 1 -8 -18 1 24 -221 -509 1 -89
\(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 1 1 -1 1 -3 1 -6 1 1 -10 8 1 -15 29
\(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 1 0 1 -2 -1 1 -5 3 2 1 -9 20 -4 -3 1
\(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ -1 -3 2 -6 16 -3 -10 58 -72 4 -15 150
\(T_{n + 2, n}\) ➤ TablDiag2 ➤ 1 -6 29 -145 814 -5243 38618 -321690
\(T_{n + 3, n}\) ➤ TablDiag3 ➤ 1 -10 75 -545 4179 -34860 318926
\(\text{lcm}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablLcm ➤ 1 1 3 24 3480 193575 12371953440
\(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 3 8 29 145 814 5243 38618 321690
\(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 1 2 9 31 165 976 6853 54797 493209
\(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 -1 -3 -7 -34 -161 -981 -6847 -54804
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 -2 1 5 -27 114 -527 2897 -18967
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 -1 -2 9 -23 55 -154 581 -2967 19055
\(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 1 0 -2 -4 -1 14 31 -11 -190 -288 708
\(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 1 -3 -6 29 75 -545 -1575 15659 47775
\(T_{2 n, n}\) ➤ CentralE ➤ 1 -3 29 -545 15659 -606417 29515079
\(T_{2 n + 1, n}\) ➤ CentralO ➤ 1 -6 75 -1575 47775 -1908060 94715621
\(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 -1 -1 7 -20 51 -149 575 -2960 19047
\(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 -1 -2 9 -23 55 -154 581 -2967 19055
\(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 -1 1 17 -94 379 -1601 8133 -50660
\(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 0 -4 6 40 -330 1096 3766 -103900
\(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 -2 8 -44 312 -2722 28128 -334616
\(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 1 2 13 28 41 46 37 8 -47 -134 -259 -428
\(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 -1 -1 13 -79 503 -3953 39317 -479071
\(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 -2 1 28 -335 3682 -47519 751552
\(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 0 -1 28 -855 36176 -2237525 199628892
\(T_{n + 1, n-k} \) ➤ RevToff11 ➤ 1 -3 1 8 -6 1 -24 29 -10 1 89 -145 75
\(T_{n + 1, n-k} \) ➤ RevTrev11 ➤ 1 1 -3 1 -6 8 1 -10 29 -24 1 -15 75
\(T^{-1}_{n + 1, n-k}\) ➤ RevTinv11 ➤ 1 3 1 10 6 1 37 31 10 1 151 160 75 15 1
\(T^{-1}_{n + 1, n-k}\) ➤ RevTrevinv11 ➤ 1 1 3 1 6 10 1 10 31 37 1 15 75 160 151
\(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 -1 1 1 -1 -3 1 8 1 -1 -24 -6 1 89 29
\(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 -1 0 1 -2 -1 -1 7 1 2 1 -23 6 -4 -3
\(T_{n + 1, n-k}\ (n-k + 1) \) ➤ RevTder ➤ 1 -3 2 8 -12 3 -24 58 -30 4 89 -290 225
\(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 -2 1 5 -27 114 -527 2897 -18967
\(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 -1 2 -4 10 -31 120 -571 3263 -21876
\(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 -1 -3 8 29 -145 -545 4179 15659
\(T_{2 n + 1, n}\) ➤ RevCentralO ➤ -1 8 -145 4179 -163191 8002742
\(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 1 -1 -1 8 -31 119 -533 2904 -18975
\(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 -2 1 5 -27 114 -527 2897 -18967
\(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 1 1 -7 18 -31 7 377 -3748 32769
\(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ -1 2 -1 -4 -1 14 47 104 191 314 479 692
\(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 2 1 -4 1 14 -47 104 -191 314 -479 692