A203010
(n-1)-st elementary symmetric function of first n Lucas numbers, starting with L(1)=1.
Original entry on oeis.org
1, 4, 19, 145, 1679, 31146, 919866, 43716030, 3345087696, 413168662224, 82432477483344, 26585428576089600, 13864587294260493504, 11694921751248976025856, 15957837208927564640940096, 35227081534568618432596098240
Offset: 1
-
f[k_] := LucasL[k]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A203010 *)
A237498
Riordan array (1/(1-x-x^2), x/(1+2*x)).
Original entry on oeis.org
1, 1, 1, 2, -1, 1, 3, 4, -3, 1, 5, -5, 10, -5, 1, 8, 15, -25, 20, -7, 1, 13, -22, 65, -65, 34, -9, 1, 21, 57, -152, 195, -133, 52, -11, 1, 34, -93, 361, -542, 461, -237, 74, -13, 1, 55, 220, -815, 1445, -1464, 935, -385, 100, -15, 1, 89, -385, 1850, -3705
Offset: 0
Triangle begins:
1;
1, 1;
2, -1, 1;
3, 4, -3, 1;
5, -5, 10, -5, 1;
8, 15, -25, 20, -7, 1;
13, -22, 65, -65, 34, -9, 1;
...
Production matrix is:
1, 1;
1, -2, 1;
2, 0, -2, 1;
4, 0, 0, -2, 1;
8, 0, 0, 0, -2, 1;
16, 0, 0, 0, 0, -2, 1;
32, 0, 0, 0, 0, 0, -2, 1;
64, 0, 0, 0, 0, 0, 0, -2, 1;
...
-
nmax=10;Flatten[CoefficientList[Series[CoefficientList[Series[(1 + 2*x) / ((1 + 2*x - y*x) * (1 - x - x^2)), {x, 0, nmax }], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 15 2017 *)
A317360
Triangle a(n, k) read by rows: coefficient triangle that gives Lucas powers and sums of Lucas powers.
Original entry on oeis.org
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
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
-
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);
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