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Expansion of 2*x*(1 - 2*x)/(1 + 2*x - 8*x^2 - sqrt(1 - 4*x^2)).
1

%I #18 Sep 08 2022 08:46:21

%S 1,1,3,8,23,64,182,512,1451,4096,11594,32768,92710,262144,741548,

%T 2097152,5931955,16777216,47454210,134217728,379628818,1073741824,

%U 3037013748,8589934592,24296051198,68719476736,194368201572,549755813888,1554944869676,4398046511104

%N Expansion of 2*x*(1 - 2*x)/(1 + 2*x - 8*x^2 - sqrt(1 - 4*x^2)).

%C Invert transform of A001405.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CentralBinomialCoefficient.html">Central Binomial Coefficient</a>

%F G.f.: 1/(1 - Sum_{k>=1} binomial(k,floor(k/2))*x^k).

%F 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 a(n) ~ 2^(3*(n-1)/2). - _Vaclav Kotesovec_, Jan 29 2020

%p a:= proc(n) option remember; `if`(n=0, 1, add(

%p a(n-i)*binomial(i, floor(i/2)), i=1..n))

%p end:

%p seq(a(n), n=0..35); # _Alois P. Heinz_, Jun 21 2018

%t 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 nmax = 29; CoefficientList[Series[1/(1 - Sum[Binomial[k, Floor[k/2]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

%t 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 (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 Cf. A001405, A026671, A054341, A075436, A293732, A293741.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jun 21 2018