|
%I #11 Nov 22 2023 22:24:25
%S 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 623,834,654,377,21,175,677,1615,2545,2859,1985,987,34,319,1411,3859,
%U 7285,9691,9451,5911,2584,55,575,2849,8855,19115,30245,35105,30407
%N 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 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 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 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 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 First ten rows:
%e 1
%e 1 3
%e 2 5 8
%e 3 13 19 21
%e 5 25 59 65 55
%e 8 50 137 231 210 144
%e 13 94 316 623 834 654 377
%e 21 175 677 1615 2545 2859 1985 987
%e 34 319 1411 3859 7285 9691 9451 5911 2584
%e 55 575 2849 8855 19115 30245 35105 30407 17345 6765
%e 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 p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - x - x^2;
%t p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y Cf. A000045 (column 1), A001906 (T(n,n-1)), A001353 (row sums, p(n,1)), A088985 (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 nonn,tabl
%O 1,3
%A _Clark Kimberling_, Nov 13 2023
|
