OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = Sum_{k=0..n} binomial(2*k-1,k)*2^(n-k)*binomial(2*n-k-1,n-k)).
a(n) ~ 2^(3*n-2) / (Gamma(3/4) * n^(1/4)) * (1 + Gamma(3/4) / (sqrt(Pi) * n^(1/4)) + Gamma(3/4)^2 / (Pi*sqrt(2*n))). - Vaclav Kotesovec, Mar 15 2015
a(n) = 2^(n-1)*binomial(2*n-1,n)*(1+hypergeom([1/2,-n],[1-2*n],2)) for n >= 2. - Peter Luschny, Mar 15 2015
D-finite with recurrence +n*(n-1)*(n-2)*a(n) -12*(n-1)*(n-2)*(2*n-3)*a(n-1) +12*(n-2)*(16*n^2-64*n+65)*a(n-2) -16*(2*n-5)*(4*n-9)*(4*n-11)*a(n-3)=0. - R. J. Mathar, Jul 27 2022
MATHEMATICA
CoefficientList[Series[(2*x)/((1-(1-8*x)^(1/4))*(1-8*x)^(3/4)), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 15 2015 *)
PROG
(Maxima)
a(n):=sum(binomial(2*k-1, k)*2^(n-k)*binomial(2*n-k-1, n-k), k, 0, n);
(Sage)
def a(n):
h = hypergeometric([1/2, -n], [1-2*n], 2).simplify() if n != 1 else 2
return 2^(n-1)*binomial(2*n-1, n)*(1+h)
[a(n) for n in range(22)] # Peter Luschny, Mar 15 2015
(PARI) x='x+O('x^50); Vec((2*x)/((1-(1-8*x)^(1/4))*(1-8*x)^(3/4))) \\ G. C. Greubel, Jun 03 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Mar 14 2015
STATUS
approved