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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 = 2*x + 1.
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%I #5 Nov 02 2014 12:18:36

%S 1,2,2,4,4,3,10,12,8,5,20,36,32,16,8,40,88,112,80,32,13,76,204,320,

%T 320,192,64,21,142,444,840,1040,864,448,128,34,260,932,2048,3040,3136,

%U 2240,1024,256,55,470,1896,4752,8176,10080,8960,5632,2304,512,89,840

%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 = 2*x + 1.

%C Sum of numbers in row n: A006190(n). Left edge: A000045 (Fibonacci numbers). Right edge: powers of 2

%e First 3 rows:

%e 1 ... 2

%e 2 ... 4 .... 4

%e 3 ... 10 ... 12 ... 8

%e First 3 polynomials: 1 + 2*x, 2 + 4*x + 4*x^2, 3 + 10*x + 12*x^2 + 8*x^3.

%t t[n_] := t[n] = Table[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[Numerator[b]]][[n]];

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

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

%Y Cf. A230000, A231730.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Nov 13 2013