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 A317360 Triangle a(n, k) read by rows: coefficient triangle that gives Lucas powers and sums of Lucas powers. 1
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS 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. EXAMPLE 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 PROG (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); CROSSREFS Cf. A000032, A000045, A055870, A010048, A027961, A005970, A305695. Sequence in context: A296461 A144696 A072248 * A177011 A092276 A011274 Adjacent sequences:  A317357 A317358 A317359 * A317361 A317362 A317363 KEYWORD sign,tabl AUTHOR Tony Foster III, Jul 26 2018 STATUS approved

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Last modified May 10 04:41 EDT 2021. Contains 343748 sequences. (Running on oeis4.)