A188267 Coefficient of x^n in the series 1/(1-x*F(1/2,1/2;1;16x)), where F(a1,a2;b;x) is the hypergeometric series.

1, 1, 5, 45, 501, 6161, 80189, 1082649, 14996021, 211674805, 3031568597, 43920006709, 642265758053, 9465144429045, 140400306506101, 2094220410467877, 31387767877371013, 472406259202624889, 7136241394473619133, 108153547914919084017

Offset

0,3

Comments

Equivalently, coefficient of x^n in the series 1/(1-(2x/Pi)*K(16x)), where K(x) = (Pi/2)*F(1/2,1/2;1;x) is the complete elliptic integral (defined as in Mathematica, i.e. with x instead of x^2).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Formula

Recurrence: a(n+1) = sum(binomial(2k,k)^2*a(n-k),k=0..n).

G.f.: 1/(1 - x/AGM(sqrt(1 - 16*x), 1)). - Vaclav Kotesovec, Sep 30 2019

a(n) ~ Pi * 2^(4*n + 4) / (n * (log(n) - 16*Pi)^2) * (1 - 2*(gamma + 4*log(2)) / (log(n) - 16*Pi) + (3*gamma^2 - Pi^2/2 + 24*gamma*log(2) + 48*log(2)^2) / (log(n) - 16*Pi)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 01 2019

Mathematica

CoefficientList[Series[1/(1-(2x/Pi)EllipticK[16x]), {x, 0, 100}], x]
a[0] = 1; Flatten[{1, Table[a[n+1] = Sum[Binomial[2*k, k]^2*a[n-k], {k, 0, n}], {n, 0, 20}]}] (* Vaclav Kotesovec, Sep 28 2019 *)

Crossrefs

Cf. A188266, A328046.

Sequence in context: A232730 A151831 A233834 * A133305 A316705 A248586

Adjacent sequences: A188264 A188265 A188266 * A188268 A188269 A188270

Keyword

nonn,easy

Author

Emanuele Munarini, Mar 30 2011

Status

approved