A113180 Expansion of 1/sqrt((1-2*x)^2-8*x^4).
1, 2, 4, 8, 20, 56, 160, 448, 1240, 3440, 9632, 27200, 77216, 219840, 627200, 1793024, 5136480, 14743232, 42390400, 122064640, 351951232, 1015990528, 2936079360, 8493340672, 24591589120, 71262291456, 206666232832, 599778166784
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Programs
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Mathematica
CoefficientList[Series[1/Sqrt[(1-2*x)^2-8*x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 23 2014 *) -
PARI
x='x+O('x^50); Vec(1/sqrt((1-2*x)^2 - 8*x^4)) \\ G. C. Greubel, Mar 17 2017
Formula
a(n) = Sum_{k=0..floor(n/2)} C(n-2k,k)*C(n-3k,k)*2^(n-3k).
D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1) - 4*(n-1)*a(n-2) + 8*(n-2)*a(n-4). - Vaclav Kotesovec, Jun 23 2014
a(n) ~ (1+sqrt(1+2*sqrt(2)))^n / (sqrt(6+5*sqrt(2)-sqrt(70+56*sqrt(2))) * sqrt(Pi*n)). - Vaclav Kotesovec, Jun 23 2014
Comments