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