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A368156 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 + x^2.

%I A368156 #11 Jan 22 2024 00:02:35
%S A368156 1,1,2,2,4,5,3,10,14,12,5,20,41,44,29,8,40,98,148,131,70,13,76,224,
%T A368156 408,497,376,169,21,142,482,1044,1542,1588,1052,408,34,260,1003,2492,
%U A368156 4351,5456,4894,2888,985,55,470,2026,5684,11359,16790,18400,14672,7813
%N A368156 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 + x^2.
%C A368156 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 A368156 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 A368156 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 + x^2.
%F A368156 p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 8*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).
%e A368156 First eight rows:
%e A368156    1
%e A368156    1    2
%e A368156    2    4    5
%e A368156    3   10   14    12
%e A368156    5   20   41    44    29
%e A368156    8   40   98   148   131    70
%e A368156   13   76  224   408   497   376   169
%e A368156   21  142  482  1044  1542  1588  1052  408
%e A368156 Row 4 represents the polynomial p(4,x) = 3 + 10*x + 14*x^2 + 12*x^3, so (T(4,k)) = (3,10,14,12), k=0..3.
%t A368156 p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 2x^2;
%t A368156 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A368156 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A368156 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A368156 Cf. A000045 (column 1); A000129, (p(n,n-1)); A007482 (row sums), (p(n,1)); A077925 (alternating row sums), (p(n,-1)); A057088, (p(n,2)); A015523, (p(n,-2)); A015568, (p(n,3)); A180250, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368155.
%K A368156 nonn,tabl
%O A368156 1,3
%A A368156 _Clark Kimberling_, Jan 20 2024