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A317360 Triangle a(n, k) read by rows: coefficient triangle that gives Lucas powers and sums of Lucas powers.

%I A317360 #33 Aug 28 2018 03:30:46
%S A317360 1,1,2,1,7,-4,1,24,-23,-8,1,76,-164,-79,16,1,235,-960,-1045,255,32,1,
%T A317360 716,-5485,-11155,5940,831,-64,1,2166,-29816,-116480,109960,32778,
%U A317360 -2687,-128,1,6527,-158252,-1143336,2024920,1029844,-176257,-8703,256,1,19628,-822291,-10851888,34850816,32711632,-9230829,-937812,28159,512
%N A317360 Triangle a(n, k) read by rows: coefficient triangle that gives Lucas powers and sums of Lucas powers.
%F A317360 a(n, k) = Sum_{j=0..k} Lucas(k+1-j)^n * A055870(n+1, j).
%F A317360 Sum_{j=0..n} a(n, n-j) * A010048(k-1+j, n) = Lucas(k)^n.
%F A317360 Sum_{j=0..n} a(n, n-j) * A305695(k-2+j, n-1) = Sum_{t=1..k} Lucas(t)^n.
%e A317360 n\k|  0  1      2       3        4       5         6        7      8     9
%e A317360 ---+-------------------------------------------------------------------------
%e A317360 0  |  1
%e A317360 1  |  1  2
%e A317360 2  |  1  7     -4
%e A317360 3  |  1  24    -23     -8
%e A317360 4  |  1  76    -164    -79       16
%e A317360 5  |  1  235   -960    -1045     255     32
%e A317360 6  |  1  716   -5485   -11155    5940    831      -64
%e A317360 7  |  1  2166  -29816  -116480   109960  32778    -2687    -128
%e A317360 8  |  1  6527  -158252 -1143336  2024920 1029844  -176257  -8703   256
%e A317360 9  |  1  19628 -822291 -10851888 4850816 32711632 -9230829 -937812 28159 512
%o A317360 (PARI) lucas(p)=2*fibonacci(p+1)-fibonacci(p);
%o A317360 S(n, k) = (-1)^floor((k+1)/2)*(prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j)));
%o A317360 T(n, k) = sum(j=0, k, lucas(k+1-j)^n * S(n+1, j));
%o A317360 tabl(m) = for (n=0, m, for (k=0, n, print1(T(n, k), ", ")); print);
%o A317360 tabl(9);
%Y A317360 Cf. A000032, A000045, A055870, A010048, A027961, A005970, A305695.
%K A317360 sign,tabl
%O A317360 0,3
%A A317360 _Tony Foster III_, Jul 26 2018