OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: ((3*x+2)*(1-sqrt(1-4*(x^3+x^2))))/(2*(x^2+x)).
a(n) ~ (3*r+2) * sqrt(3-4*r^2) * 2^(2*n+2) * r^(n+3) * (r+1)^(n+1) / (n^(3/2) * sqrt(Pi)), where r = 0.41964337760708... is the real root of the equation 4*r^2*(1+r) = 1. - Vaclav Kotesovec, Mar 22 2016
MATHEMATICA
Table[(n+1)*Sum[Binomial[k+1, n-2*k-1] * Binomial[2*k, k] / (k+1)^2, {k, 0, (n-1)/2}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 22 2016 *)
PROG
(Maxima)
makelist(coeff(taylor(((3*x+2)*(1-sqrt(1-4*(x^3+x^2))))/(2*(x^2+x)), x, 0, 15), x, n), n, 0, 15);
a(n):=(n+1)*sum((binomial(k+1, n-2*k-1)*binomial(2*k, k))/(k+1)^2, k, 0, (n-1)/2);
(PARI) x='x+O('x^200); concat(0, Vec(((3*x+2)*(1-sqrt(1-4*(x^3+x^2))))/(2*(x^2+x)))) \\ Altug Alkan, Mar 22 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Mar 22 2016
STATUS
approved