A231729 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).

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, 8, 28, 62, 90, 103, 91, 65, 37, 17, 9, 1, 1, 13, 53, 134, 226, 296, 302, 257, 183, 110, 56, 21, 11, 1, 1, 21, 97, 273, 521, 775, 915, 903, 743, 523, 319, 167

Offset: 1

Author

Clark Kimberling, Nov 13 2013

Keywords

• nonn
• tabf

Comments

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

Examples

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

Crossrefs

Cf. A230000, A000045, A231728.

Programs

Mathematica

t[n_] := t[n] = Table[(1 + x^2)/(1 + x), {k, 0, n}];
b = Table[Factor[Convergents[t[n]]], {n, 0, 10}];
p[x_, n_] := p[x, n] = Last[Expand[Denominator[b]]][[n]];
u = Table[p[x, n], {n, 1, 10}]
v = CoefficientList[u, x]; Flatten[v]