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A368157
Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 + 2*x^2.
0
1, 1, 2, 2, 4, 6, 3, 10, 16, 16, 5, 20, 46, 56, 44, 8, 40, 108, 184, 188, 120, 13, 76, 244, 496, 692, 608, 328, 21, 142, 520, 1248, 2088, 2480, 1920, 896, 34, 260, 1074, 2936, 5764, 8256, 8592, 5952, 2448, 55, 470, 2156, 6616, 14764, 24760, 31200, 28992
OFFSET
1,3
COMMENTS
Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.
LINKS
Rigoberto Flórez, Robinson A. Higuita, and Antara Mikherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers 18 (2018) 1-28.
FORMULA
p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 + 2*x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 13*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).
EXAMPLE
First eight rows:
1
1 2
2 4 6
3 10 16 16
5 20 46 56 44
8 40 108 184 188 120
13 76 244 496 692 608 328
21 142 520 1248 2088 2480 1920 896
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 16*x^2 + 16*x^3, so (T(4,k)) = (3,10,16,16), k=0..3.
MATHEMATICA
p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 2x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
CROSSREFS
Cf. A000045 (column 1); A002605, (p(n,n-1)); A030195 (row sums), (p(n,1)); A182228 (alternating row sums), (p(n,-1)); A015545, (p(n,2)); A099012, (p(n,-2)); A087567, (p(n,3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368155, A368156.
Sequence in context: A127718 A115068 A051495 * A286542 A278973 A073256
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 20 2024
STATUS
approved