A231729 - OEIS

%I #5 Nov 02 2014 12:18:36

%S 1,1,1,1,1,1,2,4,5,3,1,1,3,7,11,11,9,5,1,1,5,15,29,35,32,22,13,7,1,1,

%T 8,28,62,90,103,91,65,37,17,9,1,1,13,53,134,226,296,302,257,183,110,

%U 56,21,11,1,1,21,97,273,521,775,915,903,743,523,319,167

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

%C Sum of numbers in row n:  2*A063727(n).  Left edge:  A000045 (Fibonacci numbers).

%e First 3 rows:

%e 1 . . . 1

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

%e 2 . . . 4 . . . 5 . . . 3 . . . 1 . . . 1 . . . 1

%e First 3 polynomials:  1 + x, 1 + x + x^2 + x^3, 2 + 4*x + 5*x^2 + 3*x^3 + x^4 + x^5.

%t t[n_] := t[n] = Table[(1 + x^2)/(1 + x), {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, A000045, A231728.

%K nonn,tabf

%O 1,7

%A _Clark Kimberling_, Nov 13 2013