Laguerre
A021009 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k}\, \frac{n!}{k!} } \)
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
A000027 \(\sum_{k=0}^{1} T_{1, k}\ n^k\) ➤ PolyRow1 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
---------- \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ RevPolyRow1 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A000142 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 1 2 6 24 120 720 5040 40320 362880
A000262 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 3 13 73 501 4051 37633 394353 4596553
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 3 13 73 501 4051 37633 394353 4596553
A000290 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 1 4 9 16 25 36 49 64 81 100 121 144 169
A001040 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 1 3 10 43 225 1393 9976 81201 740785
A001563 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 4 18 96 600 4320 35280 322560 3265920
A001809 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 9 72 600 5400 52920 564480 6531840
A001810 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 16 200 2400 29400 376320 5080320
A002720 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 2 7 34 209 1546 13327 130922 1441729
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 2 7 34 209 1546 13327 130922 1441729
A008865 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 2 7 14 23 34 47 62 79 98 119 142 167
A009940 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 0 -1 -4 -15 -56 -185 -204 6209 112400
A021009 \(T_{n, k}\) ➤ Triangle ➤ 1 1 1 2 4 1 6 18 9 1 24 96 72 16 1 120
---------- \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 -1 1 2 -4 1 -6 18 -9 1 24 -96 72 -16
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 1 -1 2 -4 1 6 -18 9 -1 24 -96 72 -16
A025166 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -1 1 7 -127 1711 -23231 334391
A025167 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 3 17 139 1473 19091 291793 5129307
A056220 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 7 17 31 49 71 97 127 161 199 241 287
A084950 \(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 1 2 1 6 4 24 18 1 120 96 9 720 600 72
A087912 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 3 14 86 648 5752 58576 671568 8546432
A102631 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 2 3 8 5 6 7 32 27 10 11 24 13 14 15
A102757 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 4 31 352 5233 95836 2080999 52189096
A103194 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 6 39 292 2505 24306 263431 3154824
A105219 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 8 63 544 5225 55656 653023 8379008
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 1 8 63 544 5225 55656 653023 8379008
A144084 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 1 1 4 2 1 9 18 6 1 16 72 96 24 1 25
---------- \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 -1 1 -4 2 1 -9 18 -6 1 -16 72 -96
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -1 1 -4 2 1 -9 18 -6 1 -16 72 -96
A163102 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 2 18 72 200 450 882 1568 2592 4050 6050
A179058 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 6 96 600 2400 7350 18816 42336 86400
A216831 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 2 11 88 905 11246 162607 2668436
A277373 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 2 14 168 2840 61870 1649232 51988748
A277382 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 4 23 168 1473 14988 173007 2228544
A295382 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -1 -2 -2 8 88 592 3344 14464 2944
A295383 \(T_{2 n, n}\) ➤ CentralE ➤ 1 4 72 2400 117600 7620480 614718720
A330260 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 2 17 352 13505 830126 74717857
A331325 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 1 3 15 97 745 6571 65359 723969
A331326 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 4 19 112 801 6756 65563 717760
A343580 \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 1 4 18 72 600 2400 29400 117600
A343840 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 0 -5 22 9 -1244 14335 -79470 -586943