A188267 -id:A188267 - cp's OEIS frontend

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

Emanuele Munarini, Mar 30 2011

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).

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

Cf. A188267, A000108.

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 *)

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

A328046 G.f.: 1/2 + 1/(1 + AGM(1, sqrt(1-16*x))).

Original entry on oeis.org

1, 1, 7, 68, 763, 9276, 118656, 1572024, 21368155, 296187164, 4169180104, 59420124472, 855590919392, 12425933510200, 181787367119112, 2676258927443328, 39615617922076635, 589234154312057436, 8801406013366190952, 131964659304934491576, 1985338775295068132520

Offset: 0

Vaclav Kotesovec, Oct 03 2019

nonn

AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

Vaclav Kotesovec, Table of n, a(n) for n = 0..830
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean

Cf. A188267.

Cf. A060691, A081085, A323385.

CoefficientList[Series[1/2 + 1/(1 + (Pi*Sqrt[1 - 16*x])/(2*EllipticK[1 - 1/(1 - 16*x)])), {x, 0, 25}], x]

a(n) ~ Pi * 16^n / (n * (log(n) + Pi)^2) * (1 - (2*gamma + 8*log(2)) / (log(n) + Pi) + (3*gamma^2 + 48*log(2)^2 + 24*gamma*log(2) - Pi^2/2) / (log(n) + Pi)^2), where gamma is the Euler-Mascheroni constant A001620.

cp's OEIS Frontend

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.

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