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Coefficient of x^n in the series 1/F(-1/2,1/2;1;16x), where F(a1,a2;b;x) is the hypergeometric series.
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%I #25 Sep 28 2019 06:55:14

%S 1,4,28,240,2316,24240,269392,3135808,37869676,471189680,6008850512,

%T 78221787968,1036166807056,13931585235520,189737945839552,

%U 2613162137898752,36344513366001452,509885938301354672,7208577711881000912

%N Coefficient of x^n in the series 1/F(-1/2,1/2;1;16x), where F(a1,a2;b;x) is the hypergeometric series.

%C Equivalently, coefficient of x^n in the series 1/((2/Pi)E(16x)), where E(x) is the complete elliptic integral of the second kind (defined as in Mathematica, i.e. with x instead of x^2).

%H Vincenzo Librandi, <a href="/A188266/b188266.txt">Table of n, a(n) for n = 0..850</a>

%F Recurrence: a(n+1) = 4*sum(k=0..n, C(k)^2*(2*k+1)*a(n-k) ), where the C(n) are the Catalan numbers (A000108).

%F Conjecture: a(n) ~ Pi * 2^(4*n-3) / n^2. - _Vaclav Kotesovec_, Apr 12 2016

%t CoefficientList[Series[(Pi/2)/EllipticE[16x],{x,0,100}],x]

%t a[0] = 1; Flatten[{1, Table[a[n+1] = 4*Sum[CatalanNumber[k]^2*(2*k + 1)*a[n-k], {k, 0, n}], {n, 0, 20}]}] (* _Vaclav Kotesovec_, Sep 28 2019 *)

%Y Cf. A188267, A000108.

%K nonn,easy

%O 0,2

%A _Emanuele Munarini_, Mar 30 2011