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A231774 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 + 2). 1

%I

%S 2,1,5,6,2,9,19,13,3,29,72,69,30,5,65,213,278,182,60,8,181,682,1084,

%T 928,451,118,13,441,1975,3795,4065,2625,1023,223,21,1165,5868,13015,

%U 16590,13290,6852,2221,414,34,2929,16697,42404,63020,60435,38799,16682

%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)/(x + 2).

%C Sum of numbers in row n: A002534(n). Left edge: A006131. Right edge: A000045 (Fibonacci numbers).

%e First 3 rows:

%e 2 ... 1

%e 5 ... 6 .... 2

%e 9 ... 19 ... 13 ... 3

%e First 3 polynomials: 2 + x, 5 + 6*x + 2*x^2, 9 + 19*x + 13*x^2 + 3*x^3.

%t t[n_] := t[n] = Table[(x + 1)/(x + 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, A231775, A000045.

%K nonn,tabf

%O 1,1

%A _Clark Kimberling_, Nov 13 2013

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Last modified September 20 10:18 EDT 2021. Contains 347584 sequences. (Running on oeis4.)