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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 numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x - 1).
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%I #5 Nov 02 2014 12:18:36

%S 1,1,2,0,2,3,1,1,3,5,0,6,0,5,8,0,8,8,0,8,13,-2,19,4,19,-2,13,21,-5,33,

%T 15,15,33,-5,21,34,-12,64,12,60,12,64,-12,34,55,-25,116,20,90,90,20,

%U 116,-25,55,89,-50,213,8,210,84,210,8,213,-50,89

%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 numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x - 1).

%C Sum of numbers in row n: 2^n. Left and right edges: A000045 (Fibonacci numbers).

%e First 5 rows:

%e 1 . . . 1

%e 2 . . . 0 . . . 2

%e 3 . . . 1 . . . 1 . . . 3

%e 5 . . . 0 . . . 6 . . . 0 . . . 5

%e 8 . . . 0 . . . 8 . . . 8 . . . 0 . . . 8

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

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

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

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

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

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

%Y Cf. A230000, A000045, A231727.

%K sign,tabf

%O 1,3

%A _Clark Kimberling_, Nov 13 2013