A048784 a(n) = tau(binomial(2*n,n)), where tau = number of divisors (A000005).

1, 2, 4, 6, 8, 18, 24, 32, 48, 48, 48, 128, 96, 192, 384, 480, 384, 768, 1152, 1536, 2304, 2048, 2048, 3840, 3456, 4608, 6144, 3840, 8192, 20480, 10240, 12288, 18432, 36864, 36864, 49152, 24576, 32768, 98304, 92160, 73728, 245760, 262144

Offset: 0

Author

David Johnson-Davies

Keywords

• nonn

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
• G. V. Fedorov, Number of divisors of the central binomial coefficient, Moscow Univ. Math. Bull., Vol. 68 (2013), pp. 194-197.

Crossrefs

Cf. A000005, A000984, A002162.

Programs

Maple

A048784 := proc(n)
    numtheory[tau](binomial(2*n,n)) ;
end proc:
seq(A048784(n),n=0..30) ; # R. J. Mathar, Jul 12 2024

Mathematica

f[n_] := DivisorSigma[0, Binomial[2 n, n]]; Table[f@n, {n, 0, 42}] (* Robert G. Wilson v, Apr 08 2009 *)

PARI

fv(n,p)=my(s);while(n\=p,s+=n);s
a(n)=my(s=1);forprime(p=2,2*n,s*=fv(2*n,p)-2*fv(n,p)+1);s \\ Charles R Greathouse IV, Aug 21 2013

Formula

a(n) = A000005(A000984(n)). - Michel Marcus, Aug 21 2013

log(a(n)) = log(2) * (pi(2*n)-pi(n)) + log(2) * (n/log(n)) * Sum_{k=0..T} c_k/log(n)^k + O(n/log(n)^(T+2)) for any T >= 0, where c_k = Sum_{m>=1} Integral_{m+1/2..m+1} log(t)^m/t^2 dt. In particular for T = 0, log(a(n)) = 2 * log(2)^2 * (n/log(n)) + O(n/log(n)^2) (Fedorov, 2013). - Amiram Eldar, Dec 10 2024