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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 + 1)/(x + 2).
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%I #5 Nov 02 2014 12:18:36

%S 2,1,2,3,1,10,17,10,2,18,47,45,19,3,58,173,210,129,40,5,130,491,769,

%T 642,302,76,8,362,1545,2850,2940,1830,687,144,13,882,4391,9565,11925,

%U 9315,4671,1469,265,21,2330,12901,31898,46195,43170,26994,11294,3049,482

%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 + 1)/(x + 2).

%C Sum of numbers in row n: 3*A002534(n). Left edge: 2*A006131. Right edge: A000045 (Fibonacci numbers).

%e First 3 rows:

%e 2 .... 1

%e 2 .... 3 .... 1

%e 10 ... 17 ... 10 ... 2

%e First 3 polynomials: 2 + x, 2 + 3*x + x^2, 10 + 17*x + 10*x^2 + 2*x^3.

%t t[n_] := t[n] = Table[(x + 1)/(x + 2), {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, A231774, A000045.

%K nonn,tabf

%O 1,1

%A _Clark Kimberling_, Nov 13 2013