A127776 a(n) = ( (2^n / n!) * Product_{k=0..n-1} (4*k + 1) )^2.

1, 4, 100, 3600, 152100, 7033104, 344622096, 17582760000, 924193822500, 49701090010000, 2721631688947600, 151241747739534400, 8507348310348810000, 483459012855561960000, 27715027900230072360000, 1600820011517288979513600, 93072675982122379574532900

Offset: 0

Author

Michael Somos, Jan 14 2007

Keywords

• nonn

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

Examples

			G.f. = 1 + 4*x + 100*x^2 + 3600*x^3 + 152100*x^4 + 7033104*x^5 + ...
		

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.

Crossrefs

Cf. A002897, A004981, A091401, A186284.

Programs

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

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 *)

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

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) = A004981(n)^2. Convolution square is A002897.

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)