OFFSET
0,2
FORMULA
G.f.: (2*(1-sqrt(1-((1-sqrt(1-4*x))^2)/x))*(1/sqrt(1-4*x)+1)/2*x)/((1-sqrt(1-4*x))^2).
Conjecture D-finite with recurrence: 2*n*(n+1)*(2*n-3)*a(n) -n*(101*n^2-312*n+203)*a(n-1) +(995*n^3-5570*n^2+9567*n-4984)*a(n-2) +2*(-2393*n^3+19300*n^2-50494*n+42835)*a(n-3) +4*(2*n7)*(1408*n^2-9889*n+16690)*a(n-4) -2600*(n-5)*(2*n-7)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Jan 27 2020
a(n) ~ 5^(2*n + 1/2) / (3^(3/2) * sqrt(Pi) * n^(3/2) * 2^(2*n - 2)). - Vaclav Kotesovec, Nov 19 2021
MATHEMATICA
Table[Sum[Binomial[2k, k]/(k+1) Binomial[2n-1, n-k], {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Feb 06 2019 *)
PROG
(Maxima)
taylor((2*(1-sqrt(1-((1-sqrt(1-4*x))^2)/x))*(1/sqrt(1-4*x)+1)/2*x)/((1-sqrt(1-4*x))^2), x, 0, 27);
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Dec 03 2016
STATUS
approved