cp's OEIS Frontend

A231154 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)/(x - 1).

%I A231154 #5 Nov 02 2014 12:18:36
%S A231154 1,1,2,0,2,3,1,1,3,5,0,6,0,5,8,0,8,8,0,8,13,-2,19,4,19,-2,13,21,-5,33,
%T A231154 15,15,33,-5,21,34,-12,64,12,60,12,64,-12,34,55,-25,116,20,90,90,20,
%U A231154 116,-25,55,89,-50,213,8,210,84,210,8,213,-50,89
%N A231154 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)/(x - 1).
%C A231154 Sum of numbers in row n:  2^n.  Left and right edges:  A000045 (Fibonacci numbers).
%e A231154 First 5 rows:
%e A231154 1 . . . 1
%e A231154 2 . . . 0 . . . 2
%e A231154 3 . . . 1 . . . 1 . . . 3
%e A231154 5 . . . 0 . . . 6 . . . 0 . . . 5
%e A231154 8 . . . 0 . . . 8 . . . 8 . . . 0 . . . 8
%e A231154 First 3 polynomials:  1 + x, 2 + 2*x^2, 3 + x + x^2 + 3*x^3.
%t A231154 t[n_] := t[n] = Table[(x + 1)/(x - 1), {k, 0, n}];
%t A231154 b = Table[Factor[Convergents[t[n]]], {n, 0, 10}];
%t A231154 p[x_, n_] := p[x, n] = Last[Expand[Numerator[b]]][[n]];
%t A231154 u = Table[p[x, n], {n, 1, 10}]
%t A231154 v = CoefficientList[u, x]; Flatten[v]
%Y A231154 Cf. A230000, A000045, A231727.
%K A231154 sign,tabf
%O A231154 1,3
%A A231154 _Clark Kimberling_, Nov 13 2013