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 A098441 #42 Jan 21 2025 09:27:17
%S A098441 1,1,33,97,1729,8001,105441,627873,6989697,48363649,488206753,
%T A098441 3701949153,35289342529,283146701761,2610495177057,21695983405857,
%U A098441 196218339243777,1667338615773441,14917038493453089,128562758660255073
%N A098441 Expansion of 1/sqrt(1 - 2*x - 63*x^2).
%C A098441 Binomial transform of 1/sqrt(1-64*x^2).
%C A098441 It appears that a(n) is the coefficient of x^n in (x^2+x+16)^n. - _Joerg Arndt_, Jan 13 2011
%H A098441 Vincenzo Librandi, <a href="/A098441/b098441.txt">Table of n, a(n) for n = 0..200</a>
%H A098441 Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%H A098441 Paveł Szabłowski, <a href="https://cdm.ucalgary.ca/article/view/76214">Beta distributions whose moment sequences are related to integer sequences listed in the OEIS</a>, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See pp. 96-97.
%F A098441 E.g.f.: exp(x)*BesselI(0, 4*sqrt(4)*x).
%F A098441 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*binomial(n, k)*16^k.
%F A098441 D-finite with recurrence: n*a(n) = (2*n-1)*a(n-1) + 63*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 15 2012
%F A098441 a(n) ~ 3^(2*n+1)/(4*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 15 2012
%F A098441 a(n) = 4^n*GegenbauerC(n, -n, -1/8). - _Peter Luschny_, May 08 2016
%F A098441 a(n) = hypergeometric2F1((1 - n)/2, -n/2, 1, 64). - _G. C. Greubel_, Feb 21 2017
%p A098441 a := n -> simplify(4^n*GegenbauerC(n, -n, -1/8)):
%p A098441 seq(a(n), n=0..19); # _Peter Luschny_, May 08 2016
%t A098441 Table[SeriesCoefficient[1/Sqrt[1-2*x-63*x^2],{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 15 2012 *)
%t A098441 CoefficientList[Series[1/Sqrt[1-2x-63x^2],{x,0,30}],x] (* _Harvey P. Dale_, Jan 27 2017 *)
%t A098441 Table[Hypergeometric2F1[(1 - k)/2, -k/2, 1, 64], {k,0,50}] (* _G. C. Greubel_, Feb 21 2017 *)
%o A098441 (PARI) x='x+O('x^66); Vec(1/sqrt(1-2*x-63*x^2)) \\ _Joerg Arndt_, May 11 2013
%o A098441 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/Sqrt(1-2*x-63*x^2))); // _G. C. Greubel_, Oct 08 2018
%K A098441 easy,nonn
%O A098441 0,3
%A A098441 _Paul Barry_, Sep 07 2004