Rencontres
A008290 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k} derangements(n - k) } \)
A000007 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 0 0 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
---------- \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 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
A000023 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 2 -2 8 8 112 656 5504 49024 491264
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
A000142 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 2 6 24 120 720 5040 40320 362880
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 2 6 24 120 720 5040 40320 362880
---------- \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 2 6 24 120 720 5040 40320 362880
A000166 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 0 1 2 9 44 265 1854 14833 133496
A000217 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 1 3 6 10 15 21 28 36 45 55 66 78 91 105
A000240 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 0 3 8 45 264 1855 14832 133497
A000354 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 5 29 233 2329 27949 391285 6260561
A000387 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 0 6 20 135 924 7420 66744 667485
A000449 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 0 10 40 315 2464 22260 222480 2447445
A000522 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 5 16 65 326 1957 13700 109601
A001563 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 4 18 96 600 4320 35280 322560
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 4 18 96 600 4320 35280 322560
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 4 18 96 600 4320 35280 322560
A002522 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 1 2 5 10 17 26 37 50 65 82 101 122 145
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 2 5 10 17 26 37 50 65 82 101 122 145
A007290 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 2 8 20 40 70 112 168 240 330 440 572
A008290 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 1 0 1 2 3 0 1 9 8 6 0 1 44 45 20
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 1 0 1 2 -3 0 -1 9 -8 6 0 1 44
A010842 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 10 38 168 872 5296 37200 297856
A010843 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 5 -12 33 -78 261 -360 3681 13446
A052849 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 4 12 48 240 1440 10080 80640 725760
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 4 12 48 240 1440 10080 80640 725760
---------- \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 4 12 48 240 1440 10080 80640 725760
A055137 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 -1 0 1 -2 -3 0 1 -3 -8 -6 0 1 -4
A062119 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 2 12 72 480 3600 30240 282240
A062282 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 2 2 16 64 416 2848 22912 205952
A063083 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 0 4 8 56 304 2192 17408 156928
A098825 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 0 1 0 1 1 0 3 2 1 0 6 8 9 1 0 10 20
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 0 1 0 1 1 0 3 -2 1 0 6 -8 9 1 0 10
A174318 \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 1 3 9 45 265 1855 14833 133497
A180188 \(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 0 2 3 0 3 8 12 0 4 45 40 30 0 5 264
A217701 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 5 38 393 5144 81445 1512720
A281262 \(T_{2 n, n}\) ➤ CentralE ➤ 1 0 6 40 630 11088 244860 6362928
A343582 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 5 -3 105 -807 10413 -143595 2304081
A371995 \(T_{n - k, n - 2k} \ \ (k \le n/2)\) ➤ RevTantidiag ➤ 1 1 1 0 1 0 1 0 1 1 0 3 1 0 6 2 1 0 10
A372102 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 1 2 4 9 19 45 107 278 728 2033