0,2

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

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

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

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

(n+1)^2 a_{n+1} = (12n^2+12n+4) a_n-32n^2 a_{n-1}. - Matthijs Coster, Apr 28, 2004

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

E.g.f.: exp(4*x)*BesselI(0, 2*x)^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), 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 (vladeta(AT)eunet.rs), Sep 16 2003

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

(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)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 04 2009]

A053175(n)=a(n)*2^n. Cf. A089603.

Sequence in context: A136783 A080609 A003645 this_sequence A108447 A028475 A128327

Adjacent sequences: A081082 A081083 A081084 this_sequence A081086 A081087 A081088

nonn,easy

Michael Somos, Mar 04 2003