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A249130 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

%I A249130 #17 Feb 28 2025 07:45:23
%S A249130 1,2,1,2,2,1,8,6,2,1,8,16,10,2,1,48,44,28,16,2,1,48,144,104,40,22,2,1,
%T A249130 384,400,368,232,56,30,2,1,384,1536,1232,688,408,72,38,2,1,3840,4384,
%U A249130 5216,3552,1248,708,92,48,2,1,3840,19200,16704,12096,7632,1968,1088,112,58,2,1
%N A249130 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
%C A249130 The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + 2*floor((n+1)/2)/f(n-1,x), where f(0,x) = 1.
%C A249130 (Sum of numbers in row n) = A249131(n) for n >= 0.
%C A249130 (Column 1) = A037223.
%H A249130 Clark Kimberling, <a href="/A249130/b249130.txt">Rows 0..100, flattened</a>
%e A249130 f(0,x) = 1/1, so that p(0,x) = 1;
%e A249130 f(1,x) = (2 + x)/1, so that p(1,x) = 2 + x;
%e A249130 f(2,x) = (2 + 2*x + x^2)/(3 + x), so that p(2,x) = 2 + 2*x + x^2.
%e A249130 First 6 rows of the triangle of coefficients:
%e A249130   1
%e A249130   2    1
%e A249130   2    2    1
%e A249130   8    6    2    1
%e A249130   8    16   10   2    1
%e A249130   48   44   28   16   2   1
%t A249130 z = 15; p[x_, n_] := x + 2 Floor[n/2]/p[x, n - 1]; p[x_, 1] = 1;
%t A249130 t = Table[Factor[p[x, n]], {n, 1, z}]
%t A249130 u = Numerator[t]
%t A249130 TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249130 array *)
%t A249130 Flatten[CoefficientList[u, x]] (* A249130 sequence *)
%Y A249130 Cf. A249131, A037223, A249128.
%K A249130 nonn,tabl,easy
%O A249130 0,2
%A A249130 _Clark Kimberling_, Oct 22 2014