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

Author

Tony Foster III, Jul 26 2018

Keywords

• sign
• tabl

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
		

Crossrefs

Cf. A000032, A000045, A055870, A010048, A027961, A005970, A305695.

Programs

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);

Formula

a(n, k) = Sum_{j=0..k} Lucas(k+1-j)^n * A055870(n+1, j).

Sum_{j=0..n} a(n, n-j) * A010048(k-1+j, n) = Lucas(k)^n.

Sum_{j=0..n} a(n, n-j) * A305695(k-2+j, n-1) = Sum_{t=1..k} Lucas(t)^n.