A188266 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.

1, 4, 28, 240, 2316, 24240, 269392, 3135808, 37869676, 471189680, 6008850512, 78221787968, 1036166807056, 13931585235520, 189737945839552, 2613162137898752, 36344513366001452, 509885938301354672, 7208577711881000912

Offset: 0

Author

Emanuele Munarini, Mar 30 2011

Keywords

• nonn
• easy

Comments

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).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..850

Crossrefs

Cf. A188267, A000108.

Programs

Mathematica

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

Formula

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).

Conjecture: a(n) ~ Pi * 2^(4*n-3) / n^2. - Vaclav Kotesovec, Apr 12 2016