A261042 Generating function g(0) where g(k) = 1 - x*2*(k+1)*(k+2)/(x*2*(k+1)*(k+2) - 1/g(k+1)).

1, 4, 64, 2176, 126976, 11321344, 1431568384, 243680935936, 53725527801856, 14893509177769984, 5070334006399074304, 2079588119566033616896, 1011390382859091900891136, 575501120339508919401447424, 378784713733072451034702413824, 285539131625477547496925147693056

Offset

0,2

Comments

More generally let G(y) defined by the Taylor expansion of the continued fraction

g(y,k) = 1 - (y*x*(k+1)*(k+2)) / ((y*x*(k+1)*(k+2)) - 1/g(y,k+1)). Then

G(1/2) -> A002105, G(1) -> A000182, G(2) -> A261042, G(4) -> A253165 and G(1/8)(n) *2^(n-1+padic(n,2)) -> A002425.

Links

Table of n, a(n) for n=0..15.

Formula

a(n) = 2^(2*n)*(2*n)!*[x^(2*n)] cos(x/sqrt(2))^(-2). - Peter Luschny, Sep 03 2022

Maple

eulerCF := proc(f, len) local g, k; g := 1;
for k from len-2 by -1 to 0 do g := 1 - f(k)/(f(k)-1/g) od;
PolynomialTools:-CoefficientList(convert(series(g, x, len), polynom), x) end:
A261042_list := len -> eulerCF(k -> x*2*(k+1)*(k+2), len): A261042_list(16);
# Alternative:
ser := series(cos(x/sqrt(2))^(-2), x, 32):
seq(2^(2*n)*(2*n)!*coeff(ser, x, 2*n), n = 0..15); # Peter Luschny, Sep 03 2022

Mathematica

fracGen[f_, len_] := Module[{g, k}, g[len] = 1; For[k = len-1, k >= 0, k--, g[k] = 1-f[k]/(f[k]-1/g[k+1])]; CoefficientList[g[0] + O[x]^(len+1), x] ]; A261042list[len_] := fracGen[x*2*(#+1)*(#+2)&, len-1]; A261042list[16] (* Jean-François Alcover, Aug 08 2015, after Peter Luschny *)

Prog

(Sage)
def A261042_list(len):
    f = lambda k: x*2*(k+1)*(k+2)
    g = 1
    for k in range(len-2, -1, -1):
        g = (1-f(k)/(f(k)-1/g)).simplify_rational()
    return taylor(g, x, 0, len-1).list()
A261042_list(16)

Crossrefs

Cf. A000182, A002105, A002425, A126156 (example section), A253165.

Sequence in context: A326274 A081559 A298433 * A002454 A013043 A296741

Adjacent sequences: A261039 A261040 A261041 * A261043 A261044 A261045

Keyword

nonn,easy

Author

Peter Luschny, Aug 08 2015

Status

approved