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A249138 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

%I A249138 #14 Feb 28 2025 05:53:47
%S A249138 1,1,1,2,1,1,2,4,1,1,6,5,7,1,1,6,18,8,10,1,1,24,26,46,12,14,1,1,24,96,
%T A249138 58,86,16,18,1,1,120,154,326,118,156,21,23,1,1,120,600,444,756,198,
%U A249138 246,26,28,1,1,720,1044,2556,1152,1692,324,384,32,34,1,1
%N A249138 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
%C A249138 The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + floor((n+2)/2)/f(n-1,x), where f(0,x) = 1.
%C A249138 (Sum of numbers in row n) = A056952(n+2) for n >= 0.
%C A249138 (Column 1) is essentially A081123 (repeated factorials).
%H A249138 Clark Kimberling, <a href="/A249138/b249138.txt">Rows 0..100, flattened</a>
%e A249138 f(0,x) = 1/1, so that p(0,x) = 1;
%e A249138 f(1,x) = (1 + x)/1, so that p(1,x) = 1 + x;
%e A249138 f(2,x) = (2 + x + x^2)/(1 + x), so that p(2,x) = 2 + x + x^2.
%e A249138 First 6 rows of the triangle of coefficients:
%e A249138   1
%e A249138   1    1
%e A249138   2    1    1
%e A249138   2    4    1    1
%e A249138   6    5    7    1    1
%e A249138   6    18   8    10   1   1
%t A249138 z = 15; p[x_, n_] := x + Floor[(n+1)/2]/p[x, n - 1]; p[x_, 1] = 1;
%t A249138 t = Table[Factor[p[x, n]], {n, 1, z}]
%t A249138 u = Numerator[t]
%t A249138 TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249138 array *)
%t A249138 Flatten[CoefficientList[u, x]] (* A249138 sequence *)
%Y A249138 Cf. A056952, A081123, A249128.
%K A249138 nonn,tabl,easy
%O A249138 0,4
%A A249138 _Clark Kimberling_, Oct 22 2014