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 A084868 #37 Dec 25 2017 04:00:50
%S A084868 1,2,8,36,168,796,3800,18216,87536,421292,2029592,9784088,47187536,
%T A084868 227651352,1098523504,5301727824,25590307552,123529362124,
%U A084868 596337248024,2878947861432,13899229883024,67105641925064,323993230750672
%N A084868 Main diagonal of symmetric square table A084867, in which the antidiagonal sums (A006012) form the first row shifted left.
%C A084868 The Hankel transform (see A001906 for definition) of this sequence is A000302 (powers of 4): 1, 4, 16, 64, 256, 1024, ... - _Philippe Deléham_, Aug 17 2005
%H A084868 Vincenzo Librandi, <a href="/A084868/b084868.txt">Table of n, a(n) for n = 0..200</a>
%H A084868 Vincent Pilaud, V Pons, <a href="http://arxiv.org/abs/1606.09643">Permutrees</a>, arXiv preprint arXiv:1606.09643, 2016
%F A084868 Differential equation: (16*x^3 + 12*x^2 - 8*x + 1) * x*(d/dx)A(x) + (8x^3 - 12*x^2 + 6*x - 1) * A(x) + (8x^2 - 6*x + 1) = 0.
%F A084868 G.f.: ((1 - 4*x) + 2*x * sqrt(1 - 4*x)) / (1 - 4*x - 4*x^2). a(n) * (n-1) = a(n-1) * (8*n - 14) - a(n-2) * 12*(n-3) - a(n-3) * 8*(2*n - 5), n > 2. Hankel number wall zig-zag diagonal is A011782. - _Michael Somos_, Sep 14 2003
%F A084868 INVERT transform of A028329 (offset 1). - _Michael Somos_, Jan 05 2012
%F A084868 G.f.: (1-2*x*f(x))/(1-2*x*f(x)-2*x) where f(x) is the g.f. of A000108 (Catalan numbers). - _Philippe Deléham_, Jan 30 2012
%F A084868 a(n) ~ (1-1/sqrt(2))*(2+2*sqrt(2))^n. - _Vaclav Kotesovec_, Oct 14 2012
%F A084868 From _Peter Bala_, Feb 05 2017: (Start)
%F A084868 G.f: sqrt(1 - 4*x)/(sqrt(1 - 4*x) - 2*x) = 1/(1 - 2*x/(1 - 2*x/(1 - x/(1 - x/(1 - x/(1 - ...)))))) (continued fraction). Cf. A026671, A081696.
%F A084868 Catalan transform of A006012, that is, equals A106566*A006012, as noted by _R. J. Mathar_. (End)
%e A084868 1 + 2*x + 8*x^2 + 36*x^3 + 168*x^4 + 796*x^5 + 3800*x^6 + 18216*x^7 + ...
%p A084868 1/(1-x/(sqrt(1/4-x))): series(%,x,23): seq(coeff(%,x,n),n=0..22); # _Peter Luschny_, Feb 06 2017
%t A084868 Table[SeriesCoefficient[((1-4*x)+2*x*Sqrt[1-4*x])/(1-4*x-4*x^2),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%o A084868 (PARI) {a(n) = if( n<0, 0, polcoeff((1 - 4*x + 2*x * sqrt(1 - 4*x + x * O(x^n))) /(1 - 4*x - 4*x^2), n))} /* _Michael Somos_, Jan 05 2012 */
%Y A084868 Cf. A006012, A011782, A028329, A084867, A026671, A081696.
%K A084868 nonn,easy
%O A084868 0,2
%A A084868 _Paul D. Hanna_, Jun 10 2003, Jun 11 2003