OFFSET
0,2
COMMENTS
Cayley (1878) refers to Gauss, Werke, t. iii, p. 424 for a slightly different form of a square of a hypergeometric series being hypergeometric. - Michael Somos, Jun 25 2012
REFERENCES
A. Cayley, An Identity, Messenger of Mathematics, 7 (1878), p. 69
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 0..301
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.
FORMULA
Expansion of K(k) / (Pi/2) in powers of (k * k'/4)^2, where K(k) is the complete elliptic integral of first kind evaluated at modulus k.
Expansion of 1 / AGM( 1, (1 - 16*x)^(1/2) ) in powers of x * (1 - 16*x) where AGM() is the arithmetic-geometric mean.
G.f.: F(1/4, 1/4; 1; 64*x).
a(n) ~ 64^n / (Gamma(1/4)^2 * n^(3/2)). - Vaclav Kotesovec, Sep 08 2015
From Gheorghe Coserea, Aug 26 2016: (Start)
n^2 * a(n) = 4*(4*n-3)^2 * a(n-1), with a(0) = 1.
0 = 16*x*(x+64)*y'' + 8*(3*x+128)*y' + y, where y(x) = A(x/-4096). (End)
EXAMPLE
G.f. = 1 + 4*x + 100*x^2 + 3600*x^3 + 152100*x^4 + 7033104*x^5 + ...
MATHEMATICA
From Michael Somos, Jun 25 2012 (Start)
a[ n_] := If[ n < 0, 0, (Pochhammer[ 1/4, n] 8^n / n!)^2];
a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/4, 1/4, 1, 64 x], {x, 0, n}];
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/2, 1/2, 1/2}, {1, 1}, 64 x]^(1/2), {x, 0, n}];
(End)
Join[{1}, RecurrenceTable[{a[1] == 4, a[n] == (4 (4 n - 3)^2 a[n-1] / n^2)}, a, {n, 20}]] (* Vincenzo Librandi, Aug 26 2016 *)
PROG
(PARI) {a(n) = if( n<0, 0, prod( k=1, n, (8*k - 6) / k)^2)};
(PARI) {a(n) = my(A); if( n<1, n==0, A = x * O(x^n); polcoeff( subst( 1 / agm(1, sqrt(1 - 16*x + A) ), x, serreverse( x*(1 - 16*x) + A )), n))};
(PARI)
seq(N) = {
my(a = vector(N)); a[1] = 4;
for (n=2, N, a[n] = 4*(4*n-3)^2*a[n-1]/n^2);
concat(1, a);
};
seq(15) \\ Gheorghe Coserea, Aug 26 2016
(Magma) [1] cat [n le 1 select 4 else (4*(4*n-3)^2*Self(n-1)/ n^2): n in [1..30]]; // Vincenzo Librandi, Aug 26 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jan 14 2007
STATUS
approved