OFFSET
0,2
COMMENTS
Hankel transform is A165410.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
G.f.: 1/(1-2*x-2*x^3*c(2*x^3)) = 2/(1-4*x+sqrt(1-8*x^3)) = (1-4*x-sqrt(1-8*x^3) )/(4*x*(1-2*x-x^2)), c(x) the g.f. of A000108.
G.f.: 1/(1-2*x-2*x^3/(1-2*x^3/(1-2*x^3/(1-2*x^3/(1-... (continued fraction).
a(n) = Sum_{k=0..n} if(n<=3k, 2^k*C((n+k)/2, k)*((3*k-n)/2 + 1)(1+(-1)^(n-k))/(2*(k+1)) = Sum_{k=0..n} 2^k * A165408(n,k).
a(n) = Sum_{k=0..n+1} Pell(n-k+1)*(0^k - 2^((k-2)/2)*A000108((k-2)/3)*(1+2*cos(2*Pi*(k-2)/3))/3).
(n+1)*a(n) = 2(n+1)*a(n-1) + (n+1)*a(n-2) + 4*(2*n-7)*a(n-3) - 8(2*n-7)*a(n-4) - 4*(2*n-7)*a(n-5). - R. J. Mathar, Nov 17 2011
a(n) ~ (4+sqrt(2)) * (1+sqrt(2))^n / 8. - Vaclav Kotesovec, Feb 01 2014
MATHEMATICA
CoefficientList[Series[2/(1-4*x+Sqrt[1-8*x^3]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 2/(1-4*x+Sqrt(1-8*x^3)) )); // G. C. Greubel, Nov 10 2022
(SageMath)
def A165408(n, k): return 0 if (n>3*k) else (1+(-1)^(n-k))*(3*k-n+2)*binomial(int((n+k)/2), k)/(4*(k+1))
[A165409(n) for n in range(41)] # G. C. Greubel, Nov 10 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 17 2009
STATUS
approved