Abel
A137452
\(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ is(k = 0)\ ? \ 0^n : \binom{n-1}{k-1} (-n)^{n - k} }
\)
A000007 \(T_{n , 0}\) ➤
TablCol0 ➤ 1 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 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤
TablGcd ➤ 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(\sum_{k=0}^{1} T_{1, k}\
n^k\) ➤ PolyRow1 ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A000169 \(T_{n + 1, 1}\) ➤
TablCol1 ➤ 1 2 9 64 625 7776 117649 2097152
---------- \(\text{max}_{k=0}^{n}\ | T_{n, k}
|\) ➤ TablMax ➤ 1 1 2 9 64 625 7776 117649 2097152
A000272 \(\sum_{k=0}^{n} T_{n, k}\)
➤ TablSum ➤ 1 1 3 16 125 1296 16807 262144 4782969
----------
\(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 3 16 125 1296 16807 262144
4782969
A000312 \(\sum_{k=0}^{n} T_{n, k}\
(-1)^{k}\) ➤ AltSum ➤ 1 -1 -1 -4 -27 -256 -3125 -46656
A002378 \(T_{n + 1, n}\) ➤
TablDiag1 ➤ 0 2 6 12 20 30 42 56 72 90 110 132 156
A005408 \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2
➤ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
A005563 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2
➤ 0 3 8 15 24 35 48 63 80 99 120 143 168
A007334 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2
➤ 1 2 8 50 432 4802 65536 1062882
A016778 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3
➤ 1 16 49 100 169 256 361 484 625 784 961
A052750 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf
➤ 1 1 5 49 729 14641 371293 11390625
A052752 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3
➤ 1 1 7 100 2197 65536 2476099 113379904
A053506 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 6 48
500 6480 100842 1835008 38263752
A053507 \(T_{n +
3, 3}\) ➤ TablCol3 ➤ 1 12 150 2160 36015 688128 14880348
A059297 \(T^{-1}_{n, k}\) ➤ Tinv
➤ 1 0 1 0 -2 1 0 3 -6 1 0 -4 24 -12 1 0 5
A059298 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1
-2 1 3 -6 1 -4 24 -12 1 5 -80 90 -20
A059299
\(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 -2 0 1 -6 3 0 1 -12 24 -4 0 1
A059300 \(T^{-1}_{n + 1, n - k + 1}\)
➤ Trevinv11 ➤ 1 1 -2 1 -6 3 1 -12 24 -4 1 -20 90 -80
A061356 \(T_{n + 1, k + 1} \) ➤
Toff11 ➤ 1 2 1 9 6 1 64 48 12 1 625 500 150 20 1
A065513 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0
➤ 0 0 2 24 300 4320 72030 1376256
A085527 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf
➤ 1 1 -3 25 -343 6561 -161051 4826809
A089946 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0
➤ 0 1 4 24 200 2160 28812 458752 8503056
A137452 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 2 1 0
9 6 1 0 64 48 12 1 0 625
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1
0 -1 0 -2 1 0 -9 6 -1 0 -64 48 -12 1
A139526
\(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 2 1 6 9 1 12 48 64 1 20 150 500
625
A193678 \(\sum_{k=0}^{n} T_{n, k}\ n^k\)
➤ PolyDiag ➤ 1 1 8 108 2048 50000 1492992 52706752
A195136 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\)
➤ OddSum ➤ 0 1 2 10 76 776 9966 154400 2803256
A225465 \(T_{n + 1, k + 1}\ (k + 1) \)
➤ Tder ➤ 1 2 2 9 12 3 64 96 36 4 625 1000 450 80
A225497 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\)
➤ TransSqrs ➤ 0 1 6 42 380 4320 59682 974848 18423288
A274278 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\)
➤ EvenSum ➤ 1 0 1 6 49 520 6841 107744 1979713
A362354 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\)
➤ PolyCol3 ➤ 1 3 15 108 1029 12288 177147 3000000
A367254 \(T_{2 n, n}\) ➤ CentralE
➤ 1 2 48 2160 143360 12600000 1379524608
A367255 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum
➤ 1 2 7 40 325 3456 45619 720896 13286025
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k +
1)\) ➤ TransNat1 ➤ 1 2 7 40 325 3456 45619 720896 13286025
A367256 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \)
➤ BinConv ➤ 1 1 5 46 593 9726 192637 4457580
A367257 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \
\binom{n}{k} \) ➤ InvBinConv ➤ 1 1 -3 10 -15 -474 12565 -258572