A165203 Expansion of (1+x)*c(x)^3/(1-x*c(x)^3), c(x) the g.f. of A000108.
1, 5, 20, 81, 332, 1372, 5702, 23793, 99576, 417664, 1754866, 7383204, 31096466, 131084954, 552969854, 2334012425, 9856336324, 41639407776, 175971686398, 743888534968, 3145439344550, 13302946909338, 56272308538682
Offset: 0
Links
- Iain Fox, Table of n, a(n) for n = 0..1595 (first 201 terms from Vincenzo Librandi)
Programs
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Magma
R
:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+x)*(1-Sqrt(1-4*x))^3/(x*(8*x^2 - (1-Sqrt(1-4*x))^3)) )); // G. C. Greubel, Jul 18 2019 -
Mathematica
Rest[CoefficientList[Series[(1+x)*((1-x)*Sqrt[1-4*x]+5*x-1)/(2*(1-4*x-x^2)), {x, 0, 30}], x]] (* Vaclav Kotesovec, Feb 01 2014 *) -
PARI
first(n) = x='x+O('x^(n+1)); Vec((1+x)*((1-x)*sqrt(1-4*x)+5*x-1)/(2*(1-4*x-x^2))) \\ Iain Fox, Feb 27 2018 -
Sage
((1+x)*(1-sqrt(1-4*x))^3/(x*(8*x^2 - (1-sqrt(1-4*x))^3)) ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 18 2019
Formula
G.f. (for offset 1): (1+x)*((1-x)*sqrt(1-4*x)+5*x-1)/(2*(1-4*x-x^2)).
Conjecture: (n+1)*(5*n-31)*a(n) +(5*n^2+74*n+62)*a(n-1) +(-285*n^2+ 1072*n-757)*a(n-2) +(695*n^2-3674*n+4206)*a(n-3) +2*(45*n-74)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Dec 11 2011
a(n) ~ (18/sqrt(5)-8) * (2+sqrt(5))^(n+2). - Vaclav Kotesovec, Feb 01 2014
Comments