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

0,2

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.

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

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 *)

Prog

(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

Crossrefs

Cf. A000005, A000984, A002162.

Sequence in context: A304660 A344902 A104001 * A152491 A355557 A329738

Adjacent sequences: A048781 A048782 A048783 * A048785 A048786 A048787

Keyword

nonn

Author

David Johnson-Davies

Status

approved