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

%I A048784 #21 Dec 12 2024 09:29:31
%S A048784 1,2,4,6,8,18,24,32,48,48,48,128,96,192,384,480,384,768,1152,1536,
%T A048784 2304,2048,2048,3840,3456,4608,6144,3840,8192,20480,10240,12288,18432,
%U A048784 36864,36864,49152,24576,32768,98304,92160,73728,245760,262144
%N A048784 a(n) = tau(binomial(2*n,n)), where tau = number of divisors (A000005).
%H A048784 Charles R Greathouse IV, <a href="/A048784/b048784.txt">Table of n, a(n) for n = 0..10000</a>
%H A048784 G. V. Fedorov, <a href="https://doi.org/10.3103/S0027132213040050">Number of divisors of the central binomial coefficient</a>, Moscow Univ. Math. Bull., Vol. 68 (2013), pp. 194-197.
%F A048784 a(n) = A000005(A000984(n)). - _Michel Marcus_, Aug 21 2013
%F A048784 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
%p A048784 A048784 := proc(n)
%p A048784     numtheory[tau](binomial(2*n,n)) ;
%p A048784 end proc:
%p A048784 seq(A048784(n),n=0..30) ; # _R. J. Mathar_, Jul 12 2024
%t A048784 f[n_] := DivisorSigma[0, Binomial[2 n, n]]; Table[f@n, {n, 0, 42}] (* _Robert G. Wilson v_, Apr 08 2009 *)
%o A048784 (PARI) fv(n,p)=my(s);while(n\=p,s+=n);s
%o A048784 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
%Y A048784 Cf. A000005, A000984, A002162.
%K A048784 nonn
%O A048784 0,2
%A A048784 _David Johnson-Davies_