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 A098443 #69 Sep 01 2025 11:15:34
%S A098443 1,4,26,184,1366,10424,80996,637424,5064166,40528984,326251276,
%T A098443 2638751504,21426682876,174563719984,1426219233416,11681133293024,
%U A098443 95877105146246,788433553532824,6494463369141116,53576199709855184
%N A098443 Expansion of 1/sqrt(1-8*x-4*x^2).
%C A098443 Binomial transform of A098444. Second binomial transform of A084770. Third binomial transform of A098264.
%H A098443 Seiichi Manyama, <a href="/A098443/b098443.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%H A098443 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.
%H A098443 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.
%F A098443 E.g.f.: exp(4*x) * BesselI(0, 2*sqrt(5)*x).
%F A098443 a(n) = Sum_{k=0..floor(n/2)} binomial(n, k) * binomial(2(n-k), n) * 2^(n-2k).
%F A098443 D-finite with recurrence: n*a(n) = 4*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 15 2012
%F A098443 a(n) ~ sqrt(50+20*sqrt(5))*(4+2*sqrt(5))^n/(10*sqrt(Pi*n)). Equivalently, a(n) ~ 2^(n-1/2) * phi^(3*n + 3/2) / (5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Oct 15 2012, updated Mar 21 2024
%F A098443 G.f.: 1/(1 - 2*x*(2+x)*Q(0)), where Q(k)= 1 + (4*k+1)*x*(2+x)/(k+1 - x*(2+x)*(2*k+2)*(4*k+3)/(2*x*(2+x)*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 15 2013
%F A098443 G.f.: Q(0), where Q(k) = 1 + 2*x*(x+2)*(4*k+1)/( 2*k+1 - x*(x+2)*(2*k+1)*(4*k+3)/(x*(x+2)*(4*k+3) + (k+1)/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Sep 16 2013
%F A098443 From _Peter Bala_, Mar 16 2024: (Start)
%F A098443 a(n) = (-2*i)^n * P(n, 2*i), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial.
%F A098443 Sum_{n >= 1} (-1)^(n+1)*4^n/(n*a(n-1)*a(n)) = 2*arctan(1/2) = 2*A073000. (End)
%F A098443 From _Seiichi Manyama_, Aug 29 2025: (Start)
%F A098443 a(n) = Sum_{k=0..n} (2-i)^k * (2+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.
%F A098443 a(n) = Sum_{k=0..floor(n/2)} 5^k * 4^(n-2*k) * binomial(n,2*k) * binomial(2*k,k).
%F A098443 a(n) = [x^n] (1+4*x+5*x^2)^n. (End)
%e A098443 G.f. = 1 + 4*x + 26*x^2 + 184*x^3 + 1366*x^4 + 10424*x^5 + 80996*x^6 + ...
%t A098443 CoefficientList[Series[1/Sqrt[1 - 8*x - 4*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 15 2012, updated Mar 21 2024 *)
%o A098443 (PARI) x='x+O('x^66); Vec(1/sqrt(1-8*x-4*x^2)) \\ _Joerg Arndt_, May 11 2013
%Y A098443 Column k=2 of A386621.
%Y A098443 Cf. A006139, A126869.
%K A098443 easy,nonn,changed
%O A098443 0,2
%A A098443 _Paul Barry_, Sep 07 2004