A305561 Expansion of 2*x*(1 - 2*x)/(1 + 2*x - 8*x^2 - sqrt(1 - 4*x^2)).

1, 1, 3, 8, 23, 64, 182, 512, 1451, 4096, 11594, 32768, 92710, 262144, 741548, 2097152, 5931955, 16777216, 47454210, 134217728, 379628818, 1073741824, 3037013748, 8589934592, 24296051198, 68719476736, 194368201572, 549755813888, 1554944869676, 4398046511104

Offset: 0

Author

Ilya Gutkovskiy, Jun 21 2018

Keywords

• nonn

Comments

Invert transform of A001405.

Links

• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Central Binomial Coefficient

Crossrefs

Cf. A001405, A026671, A054341, A075436, A293732, A293741.

Programs

Magma

m:=35; R:=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

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-i)*binomial(i, floor(i/2)), i=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 21 2018

Mathematica

nmax = 29; CoefficientList[Series[2 x (1 - 2 x)/(1 + 2 x - 8 x^2 - Sqrt[1 - 4 x^2]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[Binomial[k, Floor[k/2]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
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}]

Formula

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

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

a(n) ~ 2^(3*(n-1)/2). - Vaclav Kotesovec, Jan 29 2020