OFFSET
0,2
FORMULA
a(n) = Sum_{k=0..2n} C(n+2*k-[k/2], k)*(n-[k/2])/(n+2*k-[k/2]).
G.f. A(x) satisfies: A(x^2) = ((1+x)/(2 - x*(1-sqrt(1 - 4*x))) - (1-x)/(2 + x*(1-sqrt(1 + 4*x))))/x.
a(n) ~ 5 * 2^(4*n + 1/2) / (49*sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 09 2020
MAPLE
A100248 := proc(n)
if n = 0 then
1;
else
add(binomial(n+2*k-floor(k/2), k) * (n-floor(k/2)) / (n+2*k-floor(k/2)), k=0..2*n) ;
fi;
end proc: # R. J. Mathar, May 06 2016
MATHEMATICA
CoefficientList[Series[(1 + Sqrt[x])/(2*Sqrt[x] + (-1 + Sqrt[1 - 4*Sqrt[x]])*x) + (1 - Sqrt[x])/(-2*Sqrt[x] + (-1 + Sqrt[1 + 4*Sqrt[x]])*x), {x, 0, 25}], x] (* Vaclav Kotesovec, Oct 09 2020 *)
PROG
(PARI) {a(n)=sum(k=0, 2*n, polcoeff(((1-sqrt(1-4*z+z^2*O(z^k)))/(2*z))^(n-k\2), k, z))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=if(n==0, 1, sum(k=0, 2*n, binomial(n+2*k-(k\2), k)*(n-(k\2))/(n+2*k-(k\2))))}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 09 2004
STATUS
approved