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A081085 Expansion of 1 / AGM(1, 1 - 8*x) in powers of x. 7
1, 4, 20, 112, 676, 4304, 28496, 194240, 1353508, 9593104, 68906320, 500281280, 3664176400, 27033720640, 200683238720, 1497639994112, 11227634469668, 84509490017680, 638344820152784, 4836914483890112, 36753795855173776 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre.

This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004

This is the exponential (also known as binomial) convolution of sequence A000984 (central binomial) with itself. See the V. Jovovic e.g.f. and a(n) formulae given below. - Wolfdieter Lang, Jan 13 2012

REFERENCES

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, May 24 1982, page 657.

E. Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131, 2013

FORMULA

(n+1)^2 * a(n+1) = (12*n^2+12*n+4) * a(n) - 32*n^2*a(n-1). - Matthijs Coster, Apr 28 2004

G.f.: 1 / AGM( 1, 1 - 8*x).

E.g.f.: exp(4*x)*BesselI(0, 2*x)^2. - Vladeta Jovovic, Aug 20 2003

a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*n-2*k, n-k)*binomial(2*k, k) = binomial(2*n, n)*hypergeom([ -n, -n, 1/2], [1, -n+1/2], -1). - Vladeta Jovovic, Sep 16 2003

E.g.f.: [Sum_{n>=0} binomial(2n,n)*x^n/n! ]^2. - Paul D. Hanna, Sep 04 2009

G.f.: Gaussian Hypergeometric function 2F1(1/2, 1/2; 1; 16*x-64*x^2)  - Mark van Hoeij, Oct 24 2011

a(n) = 2^(-n) * A053175(n).

a(n) ~ 2^(3*n+1)/(Pi*n). - Vaclav Kotesovec, Oct 13 2012

EXAMPLE

G.f. = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4304*x^5 + 28496*x^6 + 194240*x^7 + ...

MATHEMATICA

Table[Sum[Binomial[n, k]*Binomial[2*n-2*k, n-k]*Binomial[2*k, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 13 2012 *)

a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/2, 1/2, 1, 16 x (1 - 4 x)], {x, 0, n}]; (* Michael Somos, Oct 25 2014 *)

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ 1 / NestWhile[ {(#[[1]] + #[[2]])/2, Sqrt[#[[1]] #[[2]]]} &, {1, Series[ 1 - 8 x, {x, 0, n}]}, #[[1]] =!= #[[2]] &] // First, {x, 0, n}]]; (* Michael Somos, Oct 27 2014 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / agm( 1, 1 - 8 * x + x * O(x^n)), n))};

(PARI) {a(n) = if( n<0, 0, 4^n * sum( k=0, n\2, binomial( n, 2*k) * binomial( 2*k, k)^2 / 16^k))};

(PARI) {a(n)=n!*polcoeff(sum(k=0, n, (2*k)!*x^k/(k!)^3 +x*O(x^n))^2, n)} /* Paul D. Hanna, Sep 04 2009 */

CROSSREFS

Cf. A053175, A089603.

Sequence in context: A227726 A080609 A003645 * A212326 A192624 A209200

Adjacent sequences:  A081082 A081083 A081084 * A081086 A081087 A081088

KEYWORD

nonn,easy

AUTHOR

Michael Somos, Mar 04 2003

STATUS

approved

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Last modified March 2 07:13 EST 2015. Contains 255131 sequences.