OFFSET
0,3
FORMULA
a(n) = sum(k = 1..n, k*4^(n-k)*binomial(n/2,n-k))/n, a(0)=1.
a(n) ~ 3 * 5^(n/2-1). - Vaclav Kotesovec, Oct 31 2014
a(n) = 3 * 5^(n/2-1) if n is even and n>0 else a(n) = ((4^(n-1)* binomial(n/2, n-1)*hypergeometric([2, 1-n],[2-n/2], -1/4))/n). - Peter Luschny, Oct 31 2014
D-finite with recurrence: (-n+1)*a(n) +(-n+2)*a(n-1) +(n+11)*a(n-2) +(n+10)*a(n-3) +20*(n-4)*a(n-4) +20*(n-5)*a(n-5)=0. - R. J. Mathar, Jan 25 2020
MAPLE
# Using function CompInv from A357588.
1, CompInv(27, n -> simplify(GegenbauerC(n-1, 1-n, 3/2))); # Peter Luschny, Oct 05 2022
MATHEMATICA
CoefficientList[Series[1/(1-x*Sqrt[4*x^2+1]-2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 31 2014 *)
PROG
(Maxima)
a(n) := if n=0 then 1 else sum(k*4^(n-k)*binomial(n/2, n-k), k, 1, n)/n;
(Sage)
def a(n):
if is_odd(n):
return simplify((4^(n-1)*binomial(n/2, n-1)*hypergeometric([2, 1-n], [2-n/2], -1/4))/n)
return 3*5^(n//2-1) if n>0 else 1
[a(n) for n in (0..27)] # Peter Luschny, Oct 31 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Oct 31 2014
STATUS
approved