A360304 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.

1, 2, 22, 436, 12326, 449596, 20023548, 1051713576, 63605620998, 4352044825708, 332356758306836, 28024237688823640, 2586049127787383644, 259239588167116230872, 28054383271233786941752, 3259794623403122252542800, 404791956361004000479206342

Offset

0,2

Links

Table of n, a(n) for n=0..16.

Formula

G.f.: sqrt( Sum_{k>=0} (2*k)!/k! * (2*x)^k ).

a(n) ~ 2^(3*n - 1/2) * n^n / exp(n). - Vaclav Kotesovec, Feb 07 2023

Mathematica

nmax = 20; CoefficientList[Series[Sqrt[Sum[(2*k)!/k!*(2*x)^k, {k, 0, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 07 2023 *)

Prog

(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);
(PARI) my(N=20, x='x+O('x^N)); Vec(sqrt(sum(k=0, N, (2*k)!/k!*(2*x)^k)))

Crossrefs

Cf. A001147, A008586.

Sequence in context: A163436 A328158 A266522 * A354943 A084949 A276454

Adjacent sequences: A360301 A360302 A360303 * A360305 A360306 A360307

Keyword

nonn

Author

Seiichi Manyama, Feb 02 2023

Status

approved