This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A127776 #34 May 31 2023 21:30:53
%S A127776 1,4,100,3600,152100,7033104,344622096,17582760000,924193822500,
%T A127776 49701090010000,2721631688947600,151241747739534400,
%U A127776 8507348310348810000,483459012855561960000,27715027900230072360000,1600820011517288979513600,93072675982122379574532900
%N A127776 a(n) = ( (2^n / n!) * Product_{k=0..n-1} (4*k + 1) )^2.
%C A127776 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
%D A127776 A. Cayley, An Identity, Messenger of Mathematics, 7 (1878), p. 69
%H A127776 Gheorghe Coserea, <a href="/A127776/b127776.txt">Table of n, a(n) for n = 0..301</a>
%H A127776 Robert S. Maier, <a href="http://arxiv.org/abs/math/0611041">On Rationally Parametrized Modular Equations</a>, arXiv:math/0611041 [math.NT], 2006.
%F A127776 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.
%F A127776 Expansion of 1 / AGM( 1, (1 - 16*x)^(1/2) ) in powers of x * (1 - 16*x) where AGM() is the arithmetic-geometric mean.
%F A127776 G.f.: F(1/4, 1/4; 1; 64*x).
%F A127776 a(n) = A004981(n)^2. Convolution square is A002897.
%F A127776 a(n) ~ 64^n / (Gamma(1/4)^2 * n^(3/2)). - _Vaclav Kotesovec_, Sep 08 2015
%F A127776 From _Gheorghe Coserea_, Aug 26 2016: (Start)
%F A127776 n^2 * a(n) = 4*(4*n-3)^2 * a(n-1), with a(0) = 1.
%F A127776 0 = 16*x*(x+64)*y'' + 8*(3*x+128)*y' + y, where y(x) = A(x/-4096). (End)
%e A127776 G.f. = 1 + 4*x + 100*x^2 + 3600*x^3 + 152100*x^4 + 7033104*x^5 + ...
%t A127776 From _Michael Somos_, Jun 25 2012 (Start)
%t A127776 a[ n_] := If[ n < 0, 0, (Pochhammer[ 1/4, n] 8^n / n!)^2];
%t A127776 a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/4, 1/4, 1, 64 x], {x, 0, n}];
%t A127776 a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/2, 1/2, 1/2}, {1, 1}, 64 x]^(1/2), {x, 0, n}];
%t A127776 (End)
%t A127776 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 *)
%o A127776 (PARI) {a(n) = if( n<0, 0, prod( k=1, n, (8*k - 6) / k)^2)};
%o A127776 (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))};
%o A127776 (PARI)
%o A127776 seq(N) = {
%o A127776 my(a = vector(N)); a[1] = 4;
%o A127776 for (n=2, N, a[n] = 4*(4*n-3)^2*a[n-1]/n^2);
%o A127776 concat(1,a);
%o A127776 };
%o A127776 seq(15) \\ _Gheorghe Coserea_, Aug 26 2016
%o A127776 (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
%Y A127776 Cf. A002897, A004981, A091401, A186284.
%K A127776 nonn
%O A127776 0,2
%A A127776 _Michael Somos_, Jan 14 2007