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 A387086 #19 Aug 26 2025 07:01:36
%S A387086 1,0,2,4,16,52,188,672,2458,9052,33648,125864,473500,1789632,6791528,
%T A387086 25863568,98796096,378411332,1452886052,5590262688,21551271916,
%U A387086 83228809640,321933018272,1247062996304,4837152438556,18785529571200,73037938668632,284268423472432
%N A387086 Expansion of B(x)/sqrt(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.
%F A387086 Sum_{k=0..n} a(k) * a(n-k) = A387085(n).
%F A387086 G.f.: 1/sqrt( 4*x - 1 + 2*sqrt(1 - 4*x) ).
%F A387086 G.f.: 1/sqrt(1 - 4*x*(-1+g)) where g = 1+x*g^2 is the g.f. of A000108.
%F A387086 G.f.: g/sqrt((-2+3*g) * (2-g)) where g = 1+x*g^2 is the g.f. of A000108.
%F A387086 a(n) ~ 2^(2*n - 1/2) / (Gamma(1/4) * n^(3/4)) * (1 - Gamma(1/4)^2/(16*Pi*sqrt(2*n))). - _Vaclav Kotesovec_, Aug 20 2025
%F A387086 D-finite with recurrence 3*n*(n-1)*a(n) -2*(n-1)*(10*n-17)*a(n-1) +4*(4*n^2-24*n+29)*a(n-2) +32*(n-2)*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, Aug 26 2025
%t A387086 nmax = 30; CoefficientList[Series[Sum[Binomial[2*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 2*(Sum[Binomial[2*n, n]*x^n, {n, 0, nmax}] - 1)], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 20 2025 *)
%o A387086 (PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(4*x-1+2*sqrt(1-4*x)))
%Y A387086 Cf. A183161, A208977, A387084.
%Y A387086 Cf. A000108, A000984, A387085.
%K A387086 nonn,changed
%O A387086 0,3
%A A387086 _Seiichi Manyama_, Aug 16 2025