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Generating function g(0) where g(k) = 1 - x*2*(k+1)*(k+2)/(x*2*(k+1)*(k+2) - 1/g(k+1)).
3

%I #22 Sep 03 2022 10:42:47

%S 1,4,64,2176,126976,11321344,1431568384,243680935936,53725527801856,

%T 14893509177769984,5070334006399074304,2079588119566033616896,

%U 1011390382859091900891136,575501120339508919401447424,378784713733072451034702413824,285539131625477547496925147693056

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

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

%C 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 a(n) = 2^(2*n)*(2*n)!*[x^(2*n)] cos(x/sqrt(2))^(-2). - _Peter Luschny_, Sep 03 2022

%p eulerCF := proc(f, len) local g, k; g := 1;

%p for k from len-2 by -1 to 0 do g := 1 - f(k)/(f(k)-1/g) od;

%p PolynomialTools:-CoefficientList(convert(series(g, x, len), polynom), x) end:

%p A261042_list := len -> eulerCF(k -> x*2*(k+1)*(k+2), len): A261042_list(16);

%p # Alternative:

%p ser := series(cos(x/sqrt(2))^(-2), x, 32):

%p seq(2^(2*n)*(2*n)!*coeff(ser, x, 2*n), n = 0..15); # _Peter Luschny_, Sep 03 2022

%t 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 (Sage)

%o def A261042_list(len):

%o f = lambda k: x*2*(k+1)*(k+2)

%o g = 1

%o for k in range(len-2,-1,-1):

%o g = (1-f(k)/(f(k)-1/g)).simplify_rational()

%o return taylor(g, x, 0, len-1).list()

%o A261042_list(16)

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

%K nonn,easy

%O 0,2

%A _Peter Luschny_, Aug 08 2015