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A367210 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 5x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >=3, where u = p(2,x), v = 1 - x - x^2.

%I A367210 #7 Nov 22 2023 22:24:43
%S A367210 1,1,5,2,9,24,3,23,63,115,5,45,191,397,551,8,90,453,1381,2358,2640,13,
%T A367210 170,1044,3807,9226,13482,12649,21,317,2249,9865,28785,58513,75061,
%U A367210 60605,34,579,4695,23703,82485,202887,357567,409779,290376,55,1045,9501
%N A367210 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 5x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >=3, where u = p(2,x), v = 1 - x - x^2.
%C A367210 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 A367210 Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, <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), Paper No. A14.
%F A367210 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 + 5x, u = p(2,x), and v = 1 - x - x^2.
%F A367210 p(n,x) = k*(b^n - c^n), where k = -(1/D), b = 1/2 (1 + 5 x - D), c = 1/2 (1 + 5 x + D), where D = sqrt(5 + 6 x + 21 x^2).
%e A367210 First eight rows:
%e A367210  1
%e A367210  1    5
%e A367210  2    9    24
%e A367210  3   23    63   115
%e A367210  5   45   191   397    551
%e A367210  8   90   453  1381   2358   2640
%e A367210 13  170  1044  3807   9226  13482  12649
%e A367210 21  317  2249  9865  28785  58513  75061  60605
%e A367210 Row 4 represents the polynomial p(4,x) = 3 + 23 x + 63 x^2 + 115 x^3, so  that (T(4,k)) = (3,23,63,115), k-0..3.
%t A367210 p[1, x_] := 1; p[2, x_] := 1 + 5 x; u[x_] := p[2, x]; v[x_] := 1 - x - x^2;
%t A367210 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A367210 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A367210 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A367210 Cf. A000045 (column 1), A004254 (T(n,n-1)), A001109 (row sums p(n,1)), A001076 (alternating row sums, p(n,-1)), A094440, A367208, A367209, A367211, A367297, A367298, A367299, A367300.
%K A367210 nonn,tabl
%O A367210 1,3
%A A367210 _Clark Kimberling_, Nov 13 2023