A308559 a(n) is the numerator of the rational part of Sum_{k>=n} binomial(2*k,k-n)^(-1).

4, 1, 23, 3, -211, 6169, 1709, -24923, 3941153, 7457, -26565167, 338662421, 29719175, -5168552017, 40526745521, 50607208969, -42190362918239, 3146154503067509, 2312776975921, -1570173112141273, 27153272350852367, 473757364639811, -132365433369215539, 1183965646415001041, 63942535017037643

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..1149

Mathematics StackExchange, Sum of reciprocal binomial coefficients

Formula

Sum_{k>=n} binomial(2*k,k-n)^(-1) = int_0^1 dt (1-t)^(2*n)*(2+(2*n-1)*(1-t+t^2))/(1-t+t^2)^2.

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) = 3.

Maple

f:= proc(n) local J;
J:= int((1-t)^(2*n)*(2+(2*n-1)*(1-t+t^2))/(1-t+t^2)^2, t=0..1);
numer(subs(Pi=0, J))
end proc:
map(f, [$0..40]);

Mathematica

a[n_] := FunctionExpand[Sum[1/Binomial[2k, k-n], {k, n, Infinity}]] /. Pi -> 0 // Numerator;
a /@ Range[0, 40] (* Jean-François Alcover, Jul 31 2020 *)

Crossrefs

Cf. A309001 (denominators).

Sequence in context: A113384 A243663 A039812 * A249268 A057869 A337204

Adjacent sequences: A308556 A308557 A308558 * A308560 A308561 A308562

Keyword

sign,frac

Author

Robert Israel, Jun 07 2019

Status

approved