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 A230001 Triangular array read by rows: row n shows the coefficients of the polynomial v(n) = d(0) + d(1)*x + ... + d(k)*x^k which is the denominator of the n-th convergent of the continued fraction [1, 1/x, 1/x^2, ... ,1/x^n]. 1

%I

%S 1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,

%T 0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,2,0,1,0,1,0,0,1,1,0,0,1,0,1,0,1,

%U 0,1,1,1,1,1,2,0,2,0,2,0,1,1,1,1,0,1

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

%C In the Name section, k = n(n+1)/2 if n is even, and k = (n-1)(n+2)/2 if n is odd. For the numerator polynomials, see A230000. Conjecture: every nonnegative integer occurs infinitely many times.

%F Write the numerator polynomials as u(0), u(1), u(2), ... and the denominator polynomials as v(0), v(1), v(2),... Let p(0) = 1, q(0) = 1; p(1) = (1 + x)/x; q(1) = 1/x; p(n ) = p(n-1)/x^n + p(n-2), q(n) = q(n-1)/x^n + q(n-2). Then u(n)/v(n) = p(n)/q(n) for n>=0.

%e The first 7 rows:

%e 1 . . . . . . . . . . . . polynomial v(0) = 1

%e 1 . . . . . . . . . . . . polynomial v(1) = 1

%e 1 0 0 1 . . . . . . . . . v(2) = 1 + x^3

%e 1 0 0 1 0 1 . . . . . . . v(3) = 1 + x^3 + x^5

%e 1 0 0 1 0 1 0 1 0 0 1

%e 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1

%t t[n_] := t[n] = Table[1/x^k, {k, 0, n}];

%t b = Table[Factor[Denominator[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]

%t Flatten[v]

%Y Cf. A230000.

%K nonn,tabf

%O 0,53

%A _Clark Kimberling_, Oct 11 2013

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Last modified January 19 17:26 EST 2022. Contains 350466 sequences. (Running on oeis4.)