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Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^(n) which is the denominator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 2)/(x + 1).
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%I #5 Nov 02 2014 12:18:36

%S 1,1,2,3,1,5,11,8,2,12,34,36,17,3,29,101,141,99,35,5,70,289,499,462,

%T 242,68,8,169,807,1659,1905,1320,552,129,13,408,2212,5272,7218,6210,

%U 3438,1196,239,21,985,5977,16198,25738,26427,18183,8383,2497,436,34

%N Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^(n) which is the denominator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 2)/(x + 1).

%C Sum of numbers in row n: 2*A015521(n). Left edge: A000129. Right edge: A000045 (Fibonacci numbers).

%e First 3 rows:

%e 1 ... 1

%e 2 ... 3 ... 1

%e 5 ... 11 .. 8 .. 2

%e First 3 polynomials: 1 + x, 2 + 3*x + x^2, 5 + 11*x + 8*x^2 + 2*x^3.

%t t[n_] := t[n] = Table[(x + 2)/(x + 1), {k, 0, n}];

%t b = Table[Factor[Convergents[t[n]]], {n, 0, 10}];

%t p[x_, n_] := p[x, n] = Last[Expand[Denominator[b]]][[n]];

%t u = Table[p[x, n], {n, 1, 10}]

%t v = CoefficientList[u, x]; Flatten[v]

%Y Cf. A230000, A231732.

%K nonn,tabf

%O 1,3

%A _Clark Kimberling_, Nov 13 2013