Worpitzky
A028246 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \sum_{j=0}^{n} \text{Eulerian}(n, j) \binom{n-j}{n-k} } \)
A000007 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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
---------- \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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 \(T_{n , n}\) ➤ TablDiag0 ➤ 1 1 2 6 24 120 720 5040 40320 362880
A000169 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 2 9 64 625 7776 117649 2097152
A000225 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 3 7 15 31 63 127 255 511 1023 2047
A000384 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 1 6 15 28 45 66 91 120 153 190 231 276
A000629 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 2 6 26 150 1082 9366 94586 1091670
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 2 6 26 150 1082 9366 94586 1091670
A000670 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 1 3 13 75 541 4683 47293 545835
---------- \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 3 13 75 541 4683 47293 545835
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 3 13 75 541 4683 47293 545835 7087261
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 3 13 75 541 4683 47293 545835 7087261
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 3 13 75 541 4683 47293 545835
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 1 3 13 75 541 4683 47293 545835
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -1 3 -13 75 -541 4683 -47293 545835
A001710 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 1 3 12 60 360 2520 20160 181440 1814400
A002378 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 2 6 12 20 30 42 56 72 90 110 132 156
A005460 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 1 7 50 390 3360 31920 332640 3780000
A005461 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 1 15 180 2100 25200 317520 4233600
A009006 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 -1 0 2 0 -16 0 272 0 -7936 0 353792 0
A028243 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 2 12 50 180 602 1932 6050 18660 57002
A028244 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 6 60 390 2100 10206 46620 204630 874500
A028246 \(T_{n, k}\) ➤ Triangle ➤ 1 1 1 1 3 2 1 7 12 6 1 15 50 60 24 1 31
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 1 -1 1 -3 2 1 -7 12 -6 1 -15 50 -60
A053440 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 3 2 7 12 6 15 50 60 24 31 180 390 360
A054255 \(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 2 2 5 6 6 18 25 26 24 84 134 149
A106340 \((T_{n, n - k})^{-1}\) ➤ Tinvrev ➤ 1 -1 1 1 -3 1 -1 9 -7 1 1 -45 55 -15 1
A123227 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 3 12 66 480 4368 47712 608016 8855040
A130850 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 1 2 3 1 6 12 7 1 24 60 50 15 1 120
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 -1 2 -3 1 6 -12 7 -1 24 -60 50 -15
A185157 \(T_{2 n, n}\) ➤ CentralE ➤ 1 3 50 2100 166824 21538440 4115105280
A201339 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 3 15 111 1095 13503 199815 3449631
A201354 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 4 28 292 4060 70564 1471708 35810212
A201355 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 4 20 132 1140 12324 160020 2424132
A229046 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 1 2 4 10 28 88 304 1144 4648 20248
A343583 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 7 49 391 3601 37927 451249 5995591
A343584 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 0 -3 -10 25 574 2653 -30234 -644079
A372312 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 2 15 292 10845 653406 58018051