A309001 a(n) is the denominator of the rational part of Sum_{k>=n} binomial(2*k,k-n)^(-1).

3, 3, 6, 4, 60, 840, 2520, 3465, 360360, 11440, 2450448, 23279256, 46558512, 356948592, 2230928700, 80313433200, 2329089562800, 144403552893600, 3702655202400, 72201776446800, 1068586291412640, 763275922437600, 5215718803323600, 40777437916893600, 103511957789037600, 718702343285249700

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

Cf. A308559 (numerators).

Sequence in context: A362334 A050346 A319619 * A142149 A132119 A180681

Adjacent sequences: A308998 A308999 A309000 * A309002 A309003 A309004

Keyword

nonn,frac

Author

Robert Israel, Jul 04 2019

Status

approved