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A248809 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

%I A248809 #28 Oct 27 2014 20:50:11
%S A248809 1,2,1,3,2,1,8,7,2,1,15,18,12,2,1,48,57,30,18,2,1,105,174,141,44,25,2,
%T A248809 1,384,561,414,285,60,33,2,1,945,1950,1830,810,510,78,42,2,1,3840,
%U A248809 6555,6090,4680,1410,840,98,52,2,1,10395,25290,26685,15000,10290
%N A248809 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
%C A248809 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 A248809 (Sum of numbers in row n) = A000982(n+1) for n >= 0.
%C A248809 (Column 1) is essentially A006882 (double factorials).
%H A248809 Clark Kimberling, <a href="/A248809/b248809.txt">Rows 0..100, flattened.</a>
%e A248809 f(0,x) = 1/1, so that p(0,x) = 1.
%e A248809 f(1,x) = (2 + x)/1, so that p(1,x) = 2 + x.
%e A248809 f(2,x) = (3 + 2 x + x^2)/(2 + x), so that p(2,x) = 3 + 2 x + x^2.
%e A248809 First 6 rows of the triangle of coefficients:
%e A248809 1
%e A248809 2    1
%e A248809 3    2     1
%e A248809 8    7     2    1
%e A248809 15   18   12    2    1
%e A248809 48   57   30    18   2    1
%t A248809 z = 15; f[x_, n_] := x + (n + 1)/f[x, n - 1]; f[x_, 0] = 1;
%t A248809 t = Table[Factor[f[x, n]], {n, 0, z}]
%t A248809 u = Numerator[t]
%t A248809 TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (*A248809 array*)
%t A248809 Flatten[CoefficientList[u, x]] (*A249809 sequence*)
%o A248809 (PARI) rown(n) = if (n==0, 1, x + (n+1)/rown(n-1));
%o A248809 tabl(nn) = for (n=0, nn, print(Vecrev(numerator(rown(n))))); \\ _Michel Marcus_, Oct 25 2014
%Y A248809 Cf. A000982, A006882.
%K A248809 nonn,tabl,easy
%O A248809 0,2
%A A248809 _Clark Kimberling_, Oct 23 2014