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A112884 Number of bits required to represent binomial(2^n, 2^(n-1)). 0

%I

%S 2,3,7,14,30,61,125,252,508,1019,2043,4090,8186,16377,32761,65528,

%T 131064,262135,524279,1048566,2097142,4194293,8388597

%N Number of bits required to represent binomial(2^n, 2^(n-1)).

%F Appears to be equal to 2^n - floor(n / 2)

%F G.f.: x*(-2*x^3 + 3*x - 2)/((x - 1)^2*(2*x^2 + x - 1)) [Conjectured by Harvey P. Dale, Apr 06 2011]

%e a(2) = 3 because binomial(2^2, 2^1) in binary = 110

%t Table[IntegerLength[Binomial[2^n,2^(n-1)],2],{n,25}] (* or *)

%t CoefficientList[Series[(-2 x^3+3x-2)/((x-1)^2 (2x^2+x-1)), {x,0,25}], x] (* _Harvey P. Dale_, Apr 06 2011 *)

%o (PHP): $LastFact = gmp_init('1'); for ($i = 2; $i !== 65536; $i *= 2) { $Fact = gmp_fact($i); $Result = gmp_div_q($Fact, gmp_pow($OldFact, 2)); $LastFact = $Fact; echo gmp_strval($Result, 2).'<br>'; }

%Y a(n) represents the size of A037293 in binary - see also the central binomial coefficients: A001405.

%K easy,nonn

%O 1,1

%A Matt Erbst (matt(AT)erbst.org), Oct 04 2005

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Last modified October 21 05:06 EDT 2021. Contains 348141 sequences. (Running on oeis4.)