login
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.
3
1, 4, 28, 240, 2316, 24240, 269392, 3135808, 37869676, 471189680, 6008850512, 78221787968, 1036166807056, 13931585235520, 189737945839552, 2613162137898752, 36344513366001452, 509885938301354672, 7208577711881000912
OFFSET
0,2
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
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
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 *)
CROSSREFS
Sequence in context: A112113 A368967 A369510 * A192625 A199561 A103211
KEYWORD
nonn,easy
AUTHOR
Emanuele Munarini, Mar 30 2011
STATUS
approved