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A038003 Odd Catalan numbers: a(n) = A000108(2^n-1).

%I A038003 #64 May 22 2025 10:20:06
%S A038003 1,1,5,429,9694845,14544636039226909,
%T A038003 94295850558771979787935384946380125,
%U A038003 11311095732253345760960290897769189975961199415637572612957718759342193629
%N A038003 Odd Catalan numbers: a(n) = A000108(2^n-1).
%C A038003 The next term has 150 digits. - _Harvey P. Dale_, Feb 22 2016
%H A038003 David Wasserman, May 07 2007, <a href="/A038003/b038003.txt">Table of n, a(n) for n = 0..9</a>
%H A038003 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 A038003 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CatalanNumber.html">Catalan Number</a>
%F A038003 a(n) = binomial(2^(n+1)-2, 2^n-1)/(2^n).
%F A038003 a(n-1) = C(2^n,2^(n-1))/(2^n - 1)/2. - _Benoit Cloitre_, Aug 17 2002
%F A038003 a(n) = A000108(2^n-1). - _David Wasserman_, May 07 2007
%t A038003 Select[CatalanNumber[Range[0,300]],OddQ] (* _Harvey P. Dale_, Feb 22 2016 *)
%o A038003 (Python)
%o A038003 from __future__ import division
%o A038003 A038003_list, c, s = [1, 1], 1, 3
%o A038003 for n in range(2,10**5+1):
%o A038003     c = (c*(4*n-2))//(n+1)
%o A038003     if n == s:
%o A038003         A038003_list.append(c)
%o A038003         s = 2*s+1 # _Chai Wah Wu_, Feb 12 2015
%o A038003 (PARI) a(n) = binomial(2^(n+1)-2, 2^n-1)/(2^n); \\ _Joerg Arndt_, Nov 05 2015
%o A038003 (Magma) [Binomial(2^(n+1)-2, 2^n-1)/(2^n): n in [0..10]]; // _Vincenzo Librandi_, Nov 01 2016
%Y A038003 Cf. A000108, A094389, A119861, A119908, A120274, A120275.
%Y A038003 Intersection of A001790 and A098597. - _Dimitri Papadopoulos_, Oct 28 2016
%K A038003 nonn
%O A038003 0,3
%A A038003 _Christian G. Bower_