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A249057 Triangular array: Row n shows the coefficients of polynomials p(n,x) defined in Comments.

%I A249057 #18 Nov 25 2022 11:48:29
%S A249057 1,4,1,5,4,1,24,11,4,1,35,52,18,4,1,192,123,84,26,4,1,315,660,285,120,
%T A249057 35,4,1,1920,1545,1500,545,160,45,4,1,3465,9180,4680,2820,930,204,56,
%U A249057 4,1,23040,22005,27180,11220,4740,1470,252,68,4,1,45045,142380
%N A249057 Triangular array:  Row n shows the coefficients of polynomials p(n,x) defined in Comments.
%C A249057 The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + (n + 1)/f(n-1,x), where f(0,x) = 1.
%C A249057 Row sums give A249059(n) for n >= 1.
%C A249057 First column is A249060 (n-th term = n!! for n >= 0).
%H A249057 Clark Kimberling, <a href="/A249057/b249057.txt">Table of n, a(n) for n = 0..5049</a>
%e A249057 f(0,x) = 1/1, so that p(0,x) = 1
%e A249057 f(1,x) = (4 + x)/1, so that p(1,x) = 4 + x;
%e A249057 f(2,x) = (5 + 4 x + x^2)/(1 + x), so that p(2,x) = 5 + 4 x + x^2.
%e A249057 First 6 rows of the triangle of coefficients:
%e A249057 1
%e A249057 4    1
%e A249057 5    4     1
%e A249057 24   11    4    1
%e A249057 35   52    18   4    1
%e A249057 192  123   84   26   4   1
%t A249057 z = 12; f[x_, n_] := x + (n+3)/f[x, n - 1];
%t A249057 f[x_, 0] = 1; t = Table[Factor[f[x, n]], {n, 0, z}];
%t A249057 u = Numerator[t]; TableForm[Rest[Table[CoefficientList[u[[n]], x], {n, 0, z}]]];
%t A249057 Flatten[CoefficientList[u, x]] (* A249057 sequence *)
%o A249057 (PARI) f(n) = if (n, x + (n + 3)/f(n-1), 1);
%o A249057 row(n) = Vecrev(numerator(f(n)), 0); \\ _Michel Marcus_, Nov 25 2022
%Y A249057 Cf. A249059, A249060.
%K A249057 nonn,tabl,easy
%O A249057 0,2
%A A249057 _Clark Kimberling_, Oct 20 2014