cp's OEIS Frontend

A230000 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(k)*x^k which is the numerator of the n-th convergent of the continued fraction [1, 1/x, 1/x^2, ... ,1/x^n].

%I A230000 #7 Aug 05 2014 09:17:22
%S A230000 1,1,1,1,1,0,1,1,1,0,1,0,1,1,1,1,0,1,0,1,1,1,1,0,1,1,1,0,1,0,1,1,1,1,
%T A230000 1,2,0,1,0,1,1,1,1,0,1,0,1,1,1,1,1,2,1,2,0,2,1,1,1,1,1,0,1,1,1,0,1,0,
%U A230000 1,1,1,1,1,2,1,2,1,3,1,2,1,2,2,1,2,1
%N A230000 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(k)*x^k which is the numerator of the n-th convergent of the continued fraction [1, 1/x, 1/x^2, ... ,1/x^n].
%C A230000 In the Name section, k = n(n+1)/2.  For the denominator polynomials, see A230001.  Conjecture:  every nonnegative integer occurs infinitely many times.
%F A230000 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 A230000 The first 7 rows:
%e A230000 1 . . . . . . . . . . . . polynomial u(0) = 1
%e A230000 1 1 . . . . . . . . . . . polynomial u(1) = 1 + x
%e A230000 1 1 0 1 . . . . . . . . . u(2) = 1 + x + x^3
%e A230000 1 1 0 1 0 1 1
%e A230000 1 1 0 1 0 1 1 1 1 0 1
%e A230000 1 1 0 1 0 1 1 1 1 1 2 0 1 0 1 1
%e A230000 1 1 0 1 0 1 1 1 1 1 2 1 2 0 2 1 1 1 1 1 0 1
%t A230000 t[n_] := t[n] = Table[1/x^k, {k, 0, n}];
%t A230000 b = Table[Factor[Convergents[t[n]]], {n, 0, 10}];
%t A230000 p[x_, n_] := p[x, n] = Last[Expand[Numerator[b]]][[n]];
%t A230000 u = Table[p[x, n], {n, 1, 10}]
%t A230000 v = CoefficientList[u, x]
%t A230000 Flatten[v]
%Y A230000 Cf. A230001.
%K A230000 nonn,tabf
%O A230000 0,36
%A A230000 _Clark Kimberling_, Oct 11 2013