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A249128 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

%I A249128 #13 Feb 28 2025 07:45:28
%S A249128 1,1,1,1,1,1,2,3,1,1,2,4,5,1,1,6,11,7,8,1,1,6,18,26,10,11,1,1,24,50,
%T A249128 46,58,14,15,1,1,24,96,154,86,102,18,19,1,1,120,274,326,444,156,177,
%U A249128 23,24,1,1,120,600,1044,756,954,246,272,28,29,1,1,720,1764,2556,3708,1692,2016,384,416,34,35,1,1
%N A249128 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
%C A249128 The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + floor((n+1)/2)/f(n-1,x), where f(0,x) = 1.
%C A249128 (Sum of numbers in row n) = A056953(n) for n >= 0.
%C A249128 Column 1 consists of repeated factorials (A000142), as in A081123.
%H A249128 Clark Kimberling, <a href="/A249128/b249128.txt">Rows 0..100, flattened</a>
%e A249128 f(0,x) = 1/1, so that p(0,x) = 1;
%e A249128 f(1,x) = (1 + x)/1, so that p(1,x) = 1 + x;
%e A249128 f(2,x) = (1 + x + x^2)/(1 + x), so that p(2,x) = 1 + x + x^2.
%e A249128 First 6 rows of the triangle of coefficients:
%e A249128   1
%e A249128   1    1
%e A249128   1    1    1
%e A249128   2    3    1    1
%e A249128   2    4    5    1    1
%e A249128   6    11   7    8    1    1
%t A249128 z = 15; p[x_, n_] := x + Floor[n/2]/p[x, n - 1]; p[x_, 1] = 1;
%t A249128 t = Table[Factor[p[x, n]], {n, 1, z}]
%t A249128 u = Numerator[t]
%t A249128 TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249128 array *)
%t A249128 Flatten[CoefficientList[u, x]] (* A249128 sequence *)
%Y A249128 Cf. A056953, A000142, A081123, A249130.
%K A249128 nonn,tabl,easy
%O A249128 0,7
%A A249128 _Clark Kimberling_, Oct 22 2014