A038003 - OEIS

%I #61 Oct 19 2024 08:32:33

%S 1,1,5,429,9694845,14544636039226909,

%T 94295850558771979787935384946380125,

%U 11311095732253345760960290897769189975961199415637572612957718759342193629

%N Odd Catalan numbers: a(n) = A000108(2^n-1).

%C The next term has 150 digits. - _Harvey P. Dale_, Feb 22 2016

%H David Wasserman, May 07 2007, <a href="/A038003/b038003.txt">Table of n, a(n) for n = 0..9</a>

%H H-Y. Lin, <a href="http://www.emis.de/journals/INTEGERS/papers/l55/l55.Abstract.html">Odd Catalan Numbers modulo 2^k</a>, Integers 11 (2011) #A55

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CatalanNumber.html">Catalan Number</a>

%F a(n) = binomial(2^(n+1)-2, 2^n-1)/(2^n).

%F a(n-1) = C(2^n,2^(n-1))/(2^n - 1)/2. - _Benoit Cloitre_, Aug 17 2002

%F a(n) = A000108(2^n-1). - _David Wasserman_, May 07 2007

%t Select[CatalanNumber[Range[0,300]],OddQ] (* _Harvey P. Dale_, Feb 22 2016 *)

%o (Python)

%o from __future__ import division

%o A038003_list, c, s = [1, 1], 1, 3

%o for n in range(2,10**5+1):

%o ....c = (c*(4*n-2))//(n+1)

%o ....if n == s:

%o ........A038003_list.append(c)

%o ........s = 2*s+1 # _Chai Wah Wu_, Feb 12 2015

%o (PARI) a(n) = binomial(2^(n+1)-2, 2^n-1)/(2^n); \\ _Joerg Arndt_, Nov 05 2015

%o (Magma) [Binomial(2^(n+1)-2, 2^n-1)/(2^n): n in [0..10]]; // _Vincenzo Librandi_, Nov 01 2016

%Y Cf. A000108, A094389, A119861, A119908, A120274, A120275.

%Y Intersection of A001790 and A098597. - _Dimitri Papadopoulos_, Oct 28 2016

%K nonn

%O 0,3

%A _Christian G. Bower_