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A081085
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Expansion of 1/AGM(1,1-8x).
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4
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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.
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. [From Wolfdieter Lang, Jan 13 2012]
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REFERENCES
| Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
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LINKS
| Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, May 24 1982.MR664643
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FORMULA
| (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]
G.f.: Gaussian Hypergeometric function 2F1(1/2, 1/2; 1; 16*x-64*x^2) - Mark van Hoeij, Oct 24 2011.
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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)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 04 2009]
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CROSSREFS
| A053175(n)=a(n)*2^n. Cf. A089603.
Sequence in context: A136783 A080609 A003645 * A192624 A108447 A028475
Adjacent sequences: A081082 A081083 A081084 * A081086 A081087 A081088
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KEYWORD
| nonn,easy
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AUTHOR
| Michael Somos, Mar 04 2003
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