OFFSET
0,1
COMMENTS
The sum is a rational number plus an integer multiple of Pi/(9 sqrt(3)).
LINKS
Robert Israel, Table of n, a(n) for n = 0..1154
Mathematics StackExchange, Sum of reciprocal binomial coefficients
FORMULA
Sum_{k>=n} binomial(2*k,k-n)^(-1) = Integral_{t=0..1} (1-t)^(2*n)*(2+(2*n-1)*(1-t+t^2))/(1-t+t^2)^2 dt.
G.f. of the rational part is -(4 + x + 4*x^2)/(3*(-1 + x)*(1 + x + x^2)) - ((1 + 3*x + x^2)*log(1 - x)*x)/(2*(1 + x + x^2)^2) + 2*arctanh(sqrt(x))*(1 + x)*x^(3/2)/(1 + x + x^2)^2.
EXAMPLE
Sum_{k>= 3) binomial(2*k,k-3)^(-1) = 3/4 + 2*Pi/(9*sqrt(3)) so a(3) = 4.
MAPLE
G:= -(4 + x + 4*x^2)/(3*(-1 + x)*(1 + x + x^2)) - ((1 + 3*x + x^2)*ln(1 - x)*x)/(2*(1 + x + x^2)^2) + 2*arctanh(sqrt(x))*(1 + x)*x^(3/2)/(1 + x + x^2)^2:
S:= series(G, x, 101):
map(denom, [seq(coeff(S, x, i), i=0..100)]);
MATHEMATICA
a[n_] := FunctionExpand[Sum[1/Binomial[2k, k-n], {k, n, Infinity}]] /. Pi -> 0 // Denominator;
a /@ Range[0, 40] (* Jean-François Alcover, Jul 31 2020 *)
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Robert Israel, Jul 04 2019
STATUS
approved