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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.

%I A368157 #13 Jan 22 2024 00:02:31
%S A368157 1,1,2,2,4,6,3,10,16,16,5,20,46,56,44,8,40,108,184,188,120,13,76,244,
%T A368157 496,692,608,328,21,142,520,1248,2088,2480,1920,896,34,260,1074,2936,
%U A368157 5764,8256,8592,5952,2448,55,470,2156,6616,14764,24760,31200,28992
%N 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.
%C A368157 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 A368157 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 A368157 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.
%F A368157 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).
%e A368157 First eight rows:
%e A368157    1
%e A368157    1    2
%e A368157    2    4    6
%e A368157    3   10   16    16
%e A368157    5   20   46    56    44
%e A368157    8   40  108   184   188   120
%e A368157   13   76  244   496   692   608   328
%e A368157   21  142  520  1248  2088  2480  1920  896
%e A368157 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.
%t A368157 p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 2x^2;
%t A368157 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A368157 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A368157 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A368157 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 A368157 nonn,tabl
%O A368157 1,3
%A A368157 _Clark Kimberling_, Jan 20 2024