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A367209 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 4*x, 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 A367209 #13 Mar 28 2025 04:08:49
%S A367209 1,1,4,2,7,15,3,18,38,56,5,35,116,186,209,8,70,273,650,859,780,13,132,
%T A367209 629,1777,3366,3821,2911,21,246,1352,4600,10410,16556,16556,10864,34,
%U A367209 449,2820,11024,29770,56874,78504,70356,40545,55,810,5701,25306,78324
%N A367209 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 4*x, 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 A367209 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 A367209 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 A367209 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 + 4*x, u = p(2,x), and v = 1 - x - x^2.
%F A367209 p(n,x) = k*(b^n - c^n), where k = -(1/D), b = (1/2)*(1 + 4*x - D), c = (1/2)*(1 + 4*x + D), where D = sqrt(5 + 4*x + 12*x^2).
%e A367209 First nine rows:
%e A367209    1
%e A367209    1    4
%e A367209    2    7    15
%e A367209    3   18    38     56
%e A367209    5   35   116    186    209
%e A367209    8   70   273    650    859    780
%e A367209   13  132   629   1777   3366   3821   2911
%e A367209   21  246  1352   4600  10410  16556  16556  10864
%e A367209   34  449  2820  11024  29770  56874  78504  70356  405459
%e A367209 Row 4 represents the polynomial p(4,x) = 3 + 18*x + 38*x^2 + 56*x^3, so (T(4,k)) = (3,18,38,56), k=0..3.
%t A367209 p[1, x_] := 1; p[2, x_] := 1 + 4 x; u[x_] := p[2, x]; v[x_] := 1 - x - x^2;
%t A367209 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A367209 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A367209 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A367209 Cf. A000045 (column 1), A001353 (T(n,n-1)), A004254 (row sums, p(n,1)), A006190 (alternating row sums, p(n,-1)), A094440, A367208, A367210, A367211, A367297, A367298, A367299, A367300.
%K A367209 nonn,tabl
%O A367209 1,3
%A A367209 _Clark Kimberling_, Nov 13 2023