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%I #33 Aug 28 2018 03:30:46
%S 1,1,2,1,7,-4,1,24,-23,-8,1,76,-164,-79,16,1,235,-960,-1045,255,32,1,
%T 716,-5485,-11155,5940,831,-64,1,2166,-29816,-116480,109960,32778,
%U -2687,-128,1,6527,-158252,-1143336,2024920,1029844,-176257,-8703,256,1,19628,-822291,-10851888,34850816,32711632,-9230829,-937812,28159,512
%N Triangle a(n, k) read by rows: coefficient triangle that gives Lucas powers and sums of Lucas powers.
%F a(n, k) = Sum_{j=0..k} Lucas(k+1-j)^n * A055870(n+1, j).
%F Sum_{j=0..n} a(n, n-j) * A010048(k-1+j, n) = Lucas(k)^n.
%F Sum_{j=0..n} a(n, n-j) * A305695(k-2+j, n-1) = Sum_{t=1..k} Lucas(t)^n.
%e n\k| 0 1 2 3 4 5 6 7 8 9
%e ---+-------------------------------------------------------------------------
%e 0 | 1
%e 1 | 1 2
%e 2 | 1 7 -4
%e 3 | 1 24 -23 -8
%e 4 | 1 76 -164 -79 16
%e 5 | 1 235 -960 -1045 255 32
%e 6 | 1 716 -5485 -11155 5940 831 -64
%e 7 | 1 2166 -29816 -116480 109960 32778 -2687 -128
%e 8 | 1 6527 -158252 -1143336 2024920 1029844 -176257 -8703 256
%e 9 | 1 19628 -822291 -10851888 4850816 32711632 -9230829 -937812 28159 512
%o (PARI) lucas(p)=2*fibonacci(p+1)-fibonacci(p);
%o S(n, k) = (-1)^floor((k+1)/2)*(prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j)));
%o T(n, k) = sum(j=0, k, lucas(k+1-j)^n * S(n+1, j));
%o tabl(m) = for (n=0, m, for (k=0, n, print1(T(n, k), ", ")); print);
%o tabl(9);
%Y Cf. A000032, A000045, A055870, A010048, A027961, A005970, A305695.
%K sign,tabl
%O 0,3
%A _Tony Foster III_, Jul 26 2018
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