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 A098456 #35 Aug 30 2025 10:13:21
%S A098456 1,2,38,212,2566,20092,210524,1884136,18854854,178415852,1764019828,
%T A098456 17115907096,169100140444,1661540282456,16458178007288,
%U A098456 162887627833552,1618680238292294,16095872154638156,160435286115927044,1600771362880092472,15997473711080724916
%N A098456 Expansion of 1/sqrt(1-4*x-64*x^2).
%C A098456 Define Q(n,x)=sum{k=0..floor(n/2), binomial(n,k)binomial(2(n-k),n)x^(n-2k)}. Then a(n)=4^n*Q(n,1/4). Central coefficients of (1+2*x+17*x^2)^n.
%H A098456 Vincenzo Librandi, <a href="/A098456/b098456.txt">Table of n, a(n) for n = 0..200</a>
%H A098456 Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%F A098456 E.g.f.: exp(2x) * BesselI(0, 2*sqrt(17)*x).
%F A098456 a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(2(n-k), n)*16^k.
%F A098456 D-finite with recurrence: n*a(n) +2*(1-2*n)*a(n-1) +64*(1-n)*a(n-2)=0. - _R. J. Mathar_, Sep 26 2012
%F A098456 a(n) ~ sqrt(578+34*sqrt(17))*(2+2*sqrt(17))^n/(34*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 15 2012
%F A098456 From _Seiichi Manyama_, Aug 30 2025: (Start)
%F A098456 a(n) = Sum_{k=0..n} (1-4*i)^k * (1+4*i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.
%F A098456 a(n) = Sum_{k=0..floor(n/2)} 17^k * 2^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). (End)
%t A098456 CoefficientList[Series[1/Sqrt[1-4x-64x^2],{x,0,30}],x] (* _Harvey P. Dale_, Dec 27 2011 *)
%o A098456 (PARI) x='x+O('x^66); Vec(1/sqrt(1-4*x-64*x^2)) \\ _Joerg Arndt_, May 11 2013
%Y A098456 Cf. A084770, A098455.
%K A098456 easy,nonn,changed
%O A098456 0,2
%A A098456 _Paul Barry_, Sep 08 2004