A231730 - OEIS

A231730

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 numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = x + 1/2.

%I #9 Nov 02 2014 12:18:36

%S 1,2,5,4,4,9,22,12,8,29,56,72,32,16,65,202,232,208,80,32,181,556,924,

%T 800,560,192,64,441,1726,2964,3480,2480,1440,448,128,1165,4832,10112,

%U 12608,11680,7168,3584,1024,256,2929,14066,31632,46752,46816,36288,19712

%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 numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = x + 1/2.

%C Sum of numbers in row n: A015521(n). Left edge: A006131. Right edge: powers of 2

%e First 3 rows:

%e 1 .... 2

%e 5 .... 4 .... 4

%e 9 .... 22 ... 12 ... 8

%e First 3 polynomials: 1 + 2*x, 5 + 4*x + 4*x^2, 9 + 22*x + 12*x^2 + 8*x^3.

%t t[n_] := t[n] = Table[x + 1/2, {k, 0, n}];

%t b = Table[Factor[Convergents[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]; Flatten[v]

%Y Cf. A230000, A231731.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Nov 13 2013