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A305561
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Expansion of 2*x*(1 - 2*x)/(1 + 2*x - 8*x^2 - sqrt(1 - 4*x^2)).
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1
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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
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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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
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MAPLE
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a:= proc(n) option remember; `if`(n=0, 1, add(
a(n-i)*binomial(i, floor(i/2)), i=1..n))
end:
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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