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A249074 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

%I A249074 #13 Feb 27 2025 07:57:45
%S A249074 1,4,1,6,4,1,32,14,4,1,60,72,24,4,1,384,228,120,36,4,1,840,1392,564,
%T A249074 176,50,4,1,6144,4488,3312,1140,240,66,4,1,15120,31200,14640,6480,
%U A249074 2040,312,84,4,1,122880,104880,97440,37440,11280,3360,392,104,4,1,332640
%N A249074 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
%C A249074 The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + 2*(n+1)/f(n-1,x), where f(0,x) = 1.
%C A249074 (Sum of numbers in row n) = A249075(n) for n >= 0.
%C A249074 (n-th term of column 1) = A087299(n) for n >= 1.
%H A249074 Clark Kimberling, <a href="/A249074/b249074.txt">Rows 0..100, flattened</a>
%e A249074 f(0,x) = 1/1, so that p(0,x) = 1;
%e A249074 f(1,x) = (4 + x)/1, so that p(1,x) = 4 + x;
%e A249074 f(2,x) = (6 + 4*x + x^2)/(4 + x), so that p(2,x) = 6 + 4*x + x^2.
%e A249074 First 6 rows of the triangle of coefficients:
%e A249074   1
%e A249074   4    1
%e A249074   6    4    1
%e A249074   32   14   4    1
%e A249074   60   72   24   4    1
%e A249074   384  228  120  36   4   1
%t A249074 z = 11; p[x_, n_] := x + 2 n/p[x, n - 1]; p[x_, 1] = 1;
%t A249074 t = Table[Factor[p[x, n]], {n, 1, z}]
%t A249074 u = Numerator[t]
%t A249074 TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249074 array *)
%t A249074 Flatten[CoefficientList[u, x]] (* A249074 sequence *)
%t A249074 v = u /. x -> 1  (* A249075 *)
%t A249074 u /. x -> 0      (* A087299 *)
%Y A249074 Cf. A249057, A249075, A087299.
%K A249074 nonn,tabl,easy
%O A249074 0,2
%A A249074 _Clark Kimberling_, Oct 20 2014