A390822 Triangular array T(n, k) read by rows: polynomials for the series expansion of the iterated function F^{t}(x) = Sum_{n>=0} (1/x)^(2*n-1)*P_n(t)/n! with F^{1}(x) = (x + sqrt(x^2 + 4))/2 and F^{2}(x) = F^{1}(F^{1}(x)). Row n of the triangle give the coefficients of the polynomial P_n(t).

1, 0, 1, 0, -1, -1, 0, 3, 6, 3, 0, -14, -45, -46, -15, 0, 80, 400, 655, 440, 105, 0, -468, -3900, -9585, -10275, -5067, -945, 0, 2268, 38556, 142905, 225750, 177072, 68334, 10395, 0, -10224, -360836, -2099412, -4796855, -5482176, -3355814, -1056828, -135135

Offset

0,8

Comments

The function F^{k}(x) satisfies: Sum_{k=0..m} 1/F^{k}(1) = F^{m}(1), with F^{0}(x) = 1.

The series expansion used in this sequence has a pole at x = 0, but the definition of the function F^{t}(x) gives us trivially F^{1}(0) = 1 and F^{t}(0) = F^{t-1}(1), such the series is not needed in this point.

F^{1}(x) is also known as the metallic mean function, thus we may name F^{t}(x) the iterated metallic mean function.

Values of t > 1 in F^{t}(x) can be interpreted as generalizations of metallic ratios. The ordinary generating function of 1/P(x), where P(x) is a minimal polynomial with its greatest root in F^{t}(m), is a linear recurrence of order 2^t. Such a generalized Fibonacci sequence corresponding to F^{2}(1) is A112576.

Links

Table of n, a(n) for n=0..44.

Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.

Mathematics Stack Exchange, What is this function related with continued fractions?.

Formula

T(n, 1) = (-1)^(n+1)*A027614(n-1), for n > 0.

T(n, 1) = A179320(n)/2.

T(n, n) = (-1)^(n+1)*A001147(n-1), for n > 0.

F^{k}(1) ~ sqrt(2*k).

F^{k}(1) is the greatest real root of the k+2'th polynomial given in A147989.

F^{-1}(x) = x-(1/x) and F^{-k}(F^{k}(x)) = x.

Example

Triangle T(n, k) begins:
              1     t    t^2     t^3     t^4     t^5    t^6    t^7
[0] (1/x)^-1  1,
[1] (1/x)^ 1  0,    1,
[2] (1/x)^ 3  0,   -1,    -1,
[3] (1/x)^ 5  0,    3,     6,      3
[4] (1/x)^ 7  0,  -14,   -45,    -46,    -15
[5] (1/x)^ 9  0,   80,   400,    655,    440,    105
[6] (1/x)^11  0, -468, -3900,  -9585, -10275,  -5067,  -945
[7] (1/x)^13  0, 2268, 38556, 142905, 225750, 177072, 68334, 10395

Prog

(PARI)
f(r) = (r+sqrt(r^2+4))/2;
c(n, k) = {my(p = 1/('x+O(x^(2*n+6)))); for(m = 1, k, p = f(p)); polcoef(p, 2*n-1); }
row(n) = {my(p = polinterpolate(vector(2*(n+1), k, k-1), vector(2*(n+1), k, c(n, k-1)))); Vecrev(p*n!); }

Crossrefs

Cf. A001147, A027614, A112576, A147989, A179320.

Cf. A001813 (row sums without sign).

Cf. A000407 (row sums without sign for only the odd or the even k).

Sequence in context: A216803 A094118 A375691 * A200478 A340263 A256158

Adjacent sequences: A390819 A390820 A390821 * A390823 A390824 A390825

Keyword

sign,tabl

Author

Thomas Scheuerle, Nov 20 2025

Status

approved