cp's OEIS Frontend

A367301 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 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 - 2*x - x^2.

%I A367301 #20 Mar 28 2025 04:35:40
%S A367301 1,3,3,10,16,8,33,75,63,21,109,320,380,220,55,360,1296,1980,1620,720,
%T A367301 144,1189,5070,9459,9940,6255,2262,377,3927,19353,42615,54561,44085,
%U A367301 22635,6909,987,12970,72532,184034,277480,272854,179972,78230,20672,2584
%N A367301 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 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 - 2*x - x^2.
%C A367301 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 A367301 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 A367301 p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 3 + 3*x, u = p(2,x), and v = 1 - 2*x - x^2.
%F A367301 p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(13 + 10*x + 5*x^2)), b = (1/2) (3*x + 3 + 1/k), c = (1/2) (3*x + 3 - 1/k).
%e A367301 First eight rows:
%e A367301      1
%e A367301      3      3
%e A367301     10     16      8
%e A367301     33     75     63     21
%e A367301    109    320    380    220     55
%e A367301    360   1296   1980   1620    720    144
%e A367301   1189   5070   9459   9940   6255   2262   377
%e A367301   3927  19353  42615  54561  44085  22635  6909  987
%e A367301 Row 4 represents the polynomial p(4,x) = 33 + 75*x + 63*x^2 + 21*x^3, so (T(4,k)) = (33,75,63,21), k=0..3.
%t A367301 p[1, x_] := 1; p[2, x_] := 3 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
%t A367301 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A367301 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A367301 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A367301 Cf. A006190 (column 1); A001906 (p(n,n-1)); A154244 (row sums, p(n,1)); A077957 (alternating row sums, p(n,-1)); A190984 (p(n,2)); A006190 (signed, p(n,-2)); A154244 (p(n,-3)); A190984 (p(n,-4)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300.
%K A367301 nonn,tabl
%O A367301 1,2
%A A367301 _Clark Kimberling_, Dec 23 2023