OFFSET
0,2
COMMENTS
Hankel transform is A176288.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1/(sqrt(1-4*x)*(1-x^2*c(x))) = 2/(sqrt(1-4*x)*(2-x+x*sqrt(1-4*x))), c(x) the g.f. of A000108.
a(n) = Sum_{k=0..floor(n/2)} C(2n-3k,n-k).
D-finite with recurrence: 2*n*a(n) +(6-11*n)*a(n-1) +(13*n-16)*a(n-2) +2*(5-n)*a(n-3) +3*(2-3*n)*a(n-4) +2*(2*n-5)*a(n-5)=0. - R. J. Mathar, Nov 15 2012 [Verified with Maple's FindRE and MinimalRecurrence functions, Georg Fischer, Nov 03 2022]
a(n) ~ 2^(2*n+3) / (7*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 04 2014
MAPLE
seq( add(binomial(2*n-3*k, n-k), k=0..floor(n/2)) , n=0..25); # G. C. Greubel, Nov 25 2019
MATHEMATICA
CoefficientList[Series[2/(Sqrt[1-4*x]*(2-x+x*Sqrt[1-4*x])), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)
a[n_]:= Sum[Binomial[2*n-3*k, n-k], {k, 0, Floor[n/2]}]; Table[a[n], {n, 0, 25}] (* G. C. Greubel, Oct 19 2016 *)
PROG
(PARI) a(n) = sum(k=0, n\2, binomial(2*n-3*k, n-k)); \\ Michel Marcus, Oct 20 2016
(Magma) [ &+[Binomial(2*n-3*k, n-k): k in [0..Floor(n/2)]] : n in [0..25]]; // G. C. Greubel, Nov 25 2019
(Sage) [sum(binomial(2*n-3*k, n-k) for k in (0..floor(n/2))) for n in (0..25)] # G. C. Greubel, Nov 25 2019
(GAP) List([0..25], n-> Sum([0..Int(n/2)], k-> Binomial(2*n-3*k, n-k) )); # G. C. Greubel, Nov 25 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 14 2010
STATUS
approved