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%I #48 Jan 30 2020 21:29:15
%S 1,2,14,36,214,604,3340,9928,52582,161708,831588,2620920,13187836,
%T 42350744,209519576,682960784,3332923526,10998087884,53067486836,
%U 176924683544,845545262996,2843923177544,13479791673896,45685735967984
%N Expansion of 1 / ((1 - 4*x) * sqrt(1 + 4*x)) in powers of x.
%H G. C. Greubel, <a href="/A091520/b091520.txt">Table of n, a(n) for n = 0..500</a>
%F G.f.: 1 / ((1 - 4*x) * sqrt(1 + 4*x)).
%F D-finite with recurrence: n*a(n) = 2 * a(n-1) + 8 * (2*n - 1) * a(n-2).
%F a(n) = 4^n * Sum_{k=0, n} binomial( 2*k, k) / (-4)^k.
%F a(n) = A003148(n) * 2^n / n!. - _Michael Somos_, Mar 17 2011
%F Asymptotics: a(n) ~ 4^n / sqrt(2).
%F G.f.: y = A(x) satisfies 0 = (16*x^2 - 1) * y' + (24*x + 2) * y and 0 = y'^3 + 8 * y'^2 * y^3 + 216 * y^5 - 256 * y^7.
%F G.f.: 1/((1-4*x)*sqrt(1+4*x)) = 1/(1-4*x+2*x*(1-4*x)/G(0)) ; G(k) = 1 + x/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Dec 21 2011
%F From _Vladimir Reshetnikov_, Nov 01 2015: (Start)
%F a(n) = 2^(n-1)*(sqrt(2)*2^n + (-1)^n*(2n+1)!!*hypergeom([1,n+3/2], [n+2], -1)/(n+1)!.
%F a(n) = 2^n*(2*n+1)!!*hypergeom([-n,1/2], [3/2], 2)/n!. (End)
%F a(n+1) = 4*a(n) - (-1)^(n)*binomial(2n+2,n+1). - _G. C. Greubel_, Nov 02 2015
%e G.f. = 1 + 2*x + 14*x^2 + 36*x^3 + 214*x^4 + 604*x^5 + 3340*x^6 + 9928*x^7 + ...
%t CoefficientList[Series[1/((1-4x)Sqrt[1+4x]),{x,0,30}],x] (* _Harvey P. Dale_, Oct 14 2013 *)
%t Table[2^n (2n+1)!! Hypergeometric2F1[-n, 1/2, 3/2, 2]/n!, {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 01 2015 *)
%t RecurrenceTable[{a[n+1] == 4*a[n] - (-1)^(n)*binom[2n+2,n+1], a[0]==1}, a, {n,0, 100}] (* _G. C. Greubel_, Nov 02 2015 *)
%o (PARI) {a(n) = if( n<0, 0, 4^n * sum( k=0, n, binomial(2*k, k) / (-4)^k))};
%o (PARI) x='x+O('x^50); Vec(1/((1-4*x)*sqrt(1+4*x))) \\ _Altug Alkan_, Nov 02 2015
%K nonn
%O 0,2
%A _Michael Somos_, Jan 18 2004
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