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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)).

%I A261042 #22 Sep 03 2022 10:42:47
%S A261042 1,4,64,2176,126976,11321344,1431568384,243680935936,53725527801856,
%T A261042 14893509177769984,5070334006399074304,2079588119566033616896,
%U A261042 1011390382859091900891136,575501120339508919401447424,378784713733072451034702413824,285539131625477547496925147693056
%N 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)).
%C A261042 More generally let G(y) defined by the Taylor expansion of the continued fraction
%C A261042 g(y,k) = 1 - (y*x*(k+1)*(k+2)) / ((y*x*(k+1)*(k+2)) - 1/g(y,k+1)). Then
%C A261042 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.
%F A261042 a(n) = 2^(2*n)*(2*n)!*[x^(2*n)] cos(x/sqrt(2))^(-2). - _Peter Luschny_, Sep 03 2022
%p A261042 eulerCF := proc(f, len) local g, k; g := 1;
%p A261042 for k from len-2 by -1 to 0 do g := 1 - f(k)/(f(k)-1/g) od;
%p A261042 PolynomialTools:-CoefficientList(convert(series(g, x, len), polynom), x) end:
%p A261042 A261042_list := len -> eulerCF(k -> x*2*(k+1)*(k+2), len): A261042_list(16);
%p A261042 # Alternative:
%p A261042 ser := series(cos(x/sqrt(2))^(-2), x, 32):
%p A261042 seq(2^(2*n)*(2*n)!*coeff(ser, x, 2*n), n = 0..15); # _Peter Luschny_, Sep 03 2022
%t A261042 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_ *)
%o A261042 (Sage)
%o A261042 def A261042_list(len):
%o A261042     f = lambda k: x*2*(k+1)*(k+2)
%o A261042     g = 1
%o A261042     for k in range(len-2,-1,-1):
%o A261042         g = (1-f(k)/(f(k)-1/g)).simplify_rational()
%o A261042     return taylor(g, x, 0, len-1).list()
%o A261042 A261042_list(16)
%Y A261042 Cf. A000182, A002105, A002425, A126156 (example section), A253165.
%K A261042 nonn,easy
%O A261042 0,2
%A A261042 _Peter Luschny_, Aug 08 2015