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Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
1

%I #28 Oct 27 2014 20:50:11

%S 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 1,384,561,414,285,60,33,2,1,945,1950,1830,810,510,78,42,2,1,3840,

%U 6555,6090,4680,1410,840,98,52,2,1,10395,25290,26685,15000,10290

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

%C 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 (Sum of numbers in row n) = A000982(n+1) for n >= 0.

%C (Column 1) is essentially A006882 (double factorials).

%H Clark Kimberling, <a href="/A248809/b248809.txt">Rows 0..100, flattened.</a>

%e f(0,x) = 1/1, so that p(0,x) = 1.

%e f(1,x) = (2 + x)/1, so that p(1,x) = 2 + x.

%e f(2,x) = (3 + 2 x + x^2)/(2 + x), so that p(2,x) = 3 + 2 x + x^2.

%e First 6 rows of the triangle of coefficients:

%e 1

%e 2 1

%e 3 2 1

%e 8 7 2 1

%e 15 18 12 2 1

%e 48 57 30 18 2 1

%t z = 15; f[x_, n_] := x + (n + 1)/f[x, n - 1]; f[x_, 0] = 1;

%t t = Table[Factor[f[x, n]], {n, 0, z}]

%t u = Numerator[t]

%t TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (*A248809 array*)

%t Flatten[CoefficientList[u, x]] (*A249809 sequence*)

%o (PARI) rown(n) = if (n==0, 1, x + (n+1)/rown(n-1));

%o tabl(nn) = for (n=0, nn, print(Vecrev(numerator(rown(n))))); \\ _Michel Marcus_, Oct 25 2014

%Y Cf. A000982, A006882.

%K nonn,tabl,easy

%O 0,2

%A _Clark Kimberling_, Oct 23 2014