A317360 Triangle a(n, k) read by rows: coefficient triangle that gives Lucas powers and sums of Lucas powers.
1, 1, 2, 1, 7, -4, 1, 24, -23, -8, 1, 76, -164, -79, 16, 1, 235, -960, -1045, 255, 32, 1, 716, -5485, -11155, 5940, 831, -64, 1, 2166, -29816, -116480, 109960, 32778, -2687, -128, 1, 6527, -158252, -1143336, 2024920, 1029844, -176257, -8703, 256, 1, 19628, -822291, -10851888, 34850816, 32711632, -9230829, -937812, 28159, 512
Offset: 0
Examples
n\k| 0 1 2 3 4 5 6 7 8 9 ---+------------------------------------------------------------------------- 0 | 1 1 | 1 2 2 | 1 7 -4 3 | 1 24 -23 -8 4 | 1 76 -164 -79 16 5 | 1 235 -960 -1045 255 32 6 | 1 716 -5485 -11155 5940 831 -64 7 | 1 2166 -29816 -116480 109960 32778 -2687 -128 8 | 1 6527 -158252 -1143336 2024920 1029844 -176257 -8703 256 9 | 1 19628 -822291 -10851888 4850816 32711632 -9230829 -937812 28159 512
Programs
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PARI
lucas(p)=2*fibonacci(p+1)-fibonacci(p); S(n, k) = (-1)^floor((k+1)/2)*(prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j))); T(n, k) = sum(j=0, k, lucas(k+1-j)^n * S(n+1, j)); tabl(m) = for (n=0, m, for (k=0, n, print1(T(n, k), ", ")); print); tabl(9);