Eulersec
A119879 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{k}\, 2^{n-k}\ \text{Euler} (n-k, 1/2) } \)
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
---------- \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 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
---------- \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ 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
A000217 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ -1 -3 -6 -10 -15 -21 -28 -36 -45 -55
A000810 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 -8 -26 352 1936 -38528 -297296
A001586 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 -3 -11 57 361 -2763 -24611 250737
---------- \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 -3 -11 57 361 -2763 -24611 250737
A003701 \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 2 4 12 36 152 624 3472 18256 126752
A005563 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ -1 0 3 8 15 24 35 48 63 80 99 120 143
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 0 -3 -8 -15 -24 -35 -48 -63 -80 -99
A009006 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 0 -2 0 16 0 -272 0 7936 0 -353792 0
---------- \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 0 -2 0 16 0 -272 0 7936 0 -353792 0
---------- \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 0 2 0 -16 0 272 0 -7936 0 353792 0
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 0 -2 0 16 0 -272 0 7936 0 -353792 0
A009725 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 2 -2 -8 16 96 -272 -2176 7936 79360
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 2 -2 -8 16 96 -272 -2176 7936 79360
A009843 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 0 -3 0 25 0 -427 0 12465 0 -555731 0
A058794 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 -2 2 18 52 110 198 322 488 702 970
A080663 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 -2 -11 -26 -47 -74 -107 -146 -191
A081658 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 0 1 0 -1 1 0 -3 0 1 0 -6 0 5 1 0
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 0 1 0 -1 1 0 -3 0 1 0 -6 0 5 1 0
A109573 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 2 0 -8 0 96 0 -2176 0 79360 0
A119467 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 1 0 1 0 3 0 1 1 0 6 0 1 0 5 0 10
A119879 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 -1 0 1 0 -3 0 1 5 0 -6 0 1 0 25 0
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 -1 0 1 0 3 0 -1 5 0 -6 0 1 0 -25
A119880 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 3 2 -3 2 63 2 -1383 2 50523 2
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 3 -2 -3 -2 63 -2 -1383 -2 50523 -2
A119881 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 8 18 32 48 128 528 512 -6912 2048
A122045 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 0 -1 0 5 0 -61 0 1385 0 -50521 0
A141665 \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 0 1 1 0 3 0 1 0 6 0 1 1 0 10 0
A155457 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 1 3 1 5 1 7 1 3 1 11 1 13 1 1 1 17
A214445 \(T_{2 n, n}\) ➤ CentralE ➤ 1 0 -6 0 350 0 -56364 0 17824950 0
A302585 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 3 18 165 2000 29855 527632 10762857
A378063 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 -3 -26 1185 15376 -2749355