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 A305561 #19 Feb 16 2025 08:33:54
%S A305561 1,1,3,8,23,64,182,512,1451,4096,11594,32768,92710,262144,741548,
%T A305561 2097152,5931955,16777216,47454210,134217728,379628818,1073741824,
%U A305561 3037013748,8589934592,24296051198,68719476736,194368201572,549755813888,1554944869676,4398046511104
%N A305561 Expansion of 2*x*(1 - 2*x)/(1 + 2*x - 8*x^2 - sqrt(1 - 4*x^2)).
%C A305561 Invert transform of A001405.
%H A305561 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305561 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CentralBinomialCoefficient.html">Central Binomial Coefficient</a>
%F A305561 G.f.: 1/(1 - Sum_{k>=1} binomial(k,floor(k/2))*x^k).
%F A305561 D-finite with recurrence: n*(n+1)*a(n) +(n-1)*(n-5)*a(n-1) -12*(n-1)*(n+1)*a(n-2) -12*(n-2)*(n-5)*a(n-3) +32*(n+1)*(n-3)*a(n-4) +32*(n-4)*(n-5)*a(n-5)=0. - _R. J. Mathar_, Jan 25 2020
%F A305561 a(n) ~ 2^(3*(n-1)/2). - _Vaclav Kotesovec_, Jan 29 2020
%p A305561 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A305561 a(n-i)*binomial(i, floor(i/2)), i=1..n))
%p A305561 end:
%p A305561 seq(a(n), n=0..35); # _Alois P. Heinz_, Jun 21 2018
%t A305561 nmax = 29; CoefficientList[Series[2 x (1 - 2 x)/(1 + 2 x - 8 x^2 - Sqrt[1 - 4 x^2]), {x, 0, nmax}], x]
%t A305561 nmax = 29; CoefficientList[Series[1/(1 - Sum[Binomial[k, Floor[k/2]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%t A305561 a[0] = 1; a[n_] := a[n] = Sum[Binomial[k, Floor[k/2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]
%o A305561 (Magma) m:=35; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 2*x*(1 - 2*x)/(1 + 2*x - 8*x^2 - Sqrt(1 - 4*x^2)))); // _Vincenzo Librandi_, Jan 27 2020
%Y A305561 Cf. A001405, A026671, A054341, A075436, A293732, A293741.
%K A305561 nonn
%O A305561 0,3
%A A305561 _Ilya Gutkovskiy_, Jun 21 2018