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%I #9 Feb 07 2023 08:36:19
%S 1,2,22,436,12326,449596,20023548,1051713576,63605620998,
%T 4352044825708,332356758306836,28024237688823640,2586049127787383644,
%U 259239588167116230872,28054383271233786941752,3259794623403122252542800,404791956361004000479206342
%N Expansion of 1/sqrt(1 - 4*1*x/(1 - 4*2*x/(1 - 4*3*x/(1 - 4*4*x/(1 - 4*5*x/(1 - ...)))))), a continued fraction.
%F G.f.: sqrt( Sum_{k>=0} (2*k)!/k! * (2*x)^k ).
%F a(n) ~ 2^(3*n - 1/2) * n^n / exp(n). - _Vaclav Kotesovec_, Feb 07 2023
%t nmax = 20; CoefficientList[Series[Sqrt[Sum[(2*k)!/k!*(2*x)^k, {k, 0, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 07 2023 *)
%o (PARI) a(n) = my(A=1+x*O(x^n)); for(i=1, n, A=1-4*(n-i+1)*x/A); polcoeff(1/sqrt(A), n);
%o (PARI) my(N=20, x='x+O('x^N)); Vec(sqrt(sum(k=0, N, (2*k)!/k!*(2*x)^k)))
%Y Cf. A001147, A008586.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Feb 02 2023
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