cp's OEIS Frontend

A367208 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*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 A367208 #18 Mar 28 2025 04:08:14
%S A367208 1,1,3,2,5,8,3,13,19,21,5,25,59,65,55,8,50,137,231,210,144,13,94,316,
%T A367208 623,834,654,377,21,175,677,1615,2545,2859,1985,987,34,319,1411,3859,
%U A367208 7285,9691,9451,5911,2584,55,575,2849,8855,19115,30245,35105,30407,17345,6765
%N A367208 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*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 A367208 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 A367208 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 A367208 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 + 3*x, u = p(2,x), and v = 1 - x - x^2.
%F A367208 p(n,x) = k*(b^n - c^n), where k = -(1/D), b = (1/2)*(1 + 3*x - D), c = (1/2)*(1 + 3*x + D), where D = sqrt(5 + 2*x + 5*x^2).
%e A367208 First ten rows:
%e A367208    1
%e A367208    1    3
%e A367208    2    5     8
%e A367208    3   13    19    21
%e A367208    5   25    59    65     55
%e A367208    8   50   137   231    210    144
%e A367208   13   94   316   623    834    654    377
%e A367208   21  175   677  1615   2545   2859   1985    987
%e A367208   34  319  1411  3859   7285   9691   9451   5911   2584
%e A367208   55  575  2849  8855  19115  30245  35105  30407  17345  6765
%e A367208 Row 4 represents the polynomial p(4,x) = 3 + 13*x + 19*x^2 + 21*x^3, so (T(4,k)) = (3,13,19,21), k=0..3.
%t A367208 p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - x - x^2;
%t A367208 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A367208 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A367208 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A367208 Cf. A000045 (column 1), A001906 (T(n,n-1)), A001353 (row sums, p(n,1)), A077985 (alternating row sums, p(n,-1)), A190974 (p(n,2)), A004254 (p(n,-2)), A190977 (p(n,-3)), A094440, A367209, A367210, A367211, A367297, A367298, A367299, A367300.
%K A367208 nonn,tabl
%O A367208 1,3
%A A367208 _Clark Kimberling_, Nov 13 2023