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A368518 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 + 3*x^2.

%I A368518 #8 Jan 25 2024 08:07:31
%S A368518 1,1,2,2,4,7,3,10,18,20,5,20,51,68,61,8,40,118,220,251,182,13,76,264,
%T A368518 584,905,888,547,21,142,558,1452,2678,3540,3076,1640,34,260,1145,3380,
%U A368518 7279,11536,13418,10456,4921,55,470,2286,7548,18391,33990,47600,49552
%N A368518 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 + 3*x^2.
%C A368518 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.
%H A368518 Rigoberto Flórez, Robinson A. Higuita, and Antara Mikherjee, <a href="http://math.colgate.edu/~integers/s14/s14.Abstract.html">Characterization of the strong divisibility property for generalized Fibonacci polynomials</a>, Integers 18 (2018) 1-28.
%F A368518 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 + 32*x^2.
%F A368518 p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 16*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).
%e A368518 First eight rows:
%e A368518    1
%e A368518    1    2
%e A368518    2    4    7
%e A368518    3   10   18    20
%e A368518    5   20   51    68    61
%e A368518    8   40  118   220   251   182
%e A368518   13   76  264   584   905   888   547
%e A368518   21  142  558  1452  2678  3540  3076  1640
%e A368518 Row 4 represents the polynomial p(4,x) = 3 + 10*x + 18*x^2 + 20*x^3, so (T(4,k)) = (3,10,18,20), k=0..3.
%t A368518 p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 3x^2;
%t A368518 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A368518 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A368518 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A368518 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.
%K A368518 nonn,tabl
%O A368518 1,3
%A A368518 _Clark Kimberling_, Jan 22 2024