%I A048881
%S A048881 0,0,1,0,1,1,2,0,1,1,2,1,2,2,3,0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,0,1,1,2,
%T A048881 1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,0,1,1,2,1,2,2,
%U A048881 3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,1,2,2,3,2,3,3,4,2,3
%N A048881 a(n) = A000120(n+1) - 1 = wt(n+1) - 1.
%C A048881 Highest power of 2 dividing n-th Catalan number (A000108).
%C A048881 a(n) = 0 iff n = 2^k - 1, k=0,1,...
%C A048881 Appears to be number of binary left-rotations (iterations of A006257) 
               to reach fixed point of form 2^k-1. Right-rotation analogue is A063250. 
               This would imply that for n >= 0, a(n)=f(n), recursively defined 
               to be 0 for n=0, otherwise as f( ( (1-n)(1-p)(1-s) - (1-n-p-s) ) 
               / 2) + p (s+1) / 2, where p = n mod 2 and s = - signum(n) (f(n<0) 
               is A000120(-n)). - Marc LeBrun (mlb(AT)well.com), Jul 11 2001.
%C A048881 Let f(0) = 01, f(1) = 12, f(2) = 23, f(3) = 34, f(4) = 45, etc. Sequence 
               gives concatenation of 0, f(0), f(f(0)), f(f(f(0))), ... Also f(f(...f(0)...)) 
               converges to A000120. - DELEHAM Philippe(kolotoko(AT)wanadoo.fr), 
               Aug 14, 2003.
%C A048881 Highest power of 2 dividing binomial(n,floor(n/2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Oct 20 2003
%C A048881 2^a(n) are numerators in the maclaurin series for (sin x)^2. [From Jacob 
               A. Siehler (siehlerj(AT)wlu.edu), Nov 11 2009]
%D A048881 R. Alter and K. K. Kubota, Prime and Prime Power Divisibility of Catalan 
               Numbers, J. Com. Th. A 15 (1973) pp. 243-256.
%F A048881 Writing n as 2^m+k with -1<=k<2^m-1, then a(n)=A000120(k+1) - Henry Bottomley 
               (se16(AT)btinternet.com), Mar 28 2000
%F A048881 a(n)= k if 2^k divides A000108(n) but 2^(k+1) does not divide A000108(n).
%F A048881 a(2*n) = a(n-1)+1, a(2*n+1) = a(n). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Oct 10 2002
%F A048881 G.f.: 1/(x-x^2) * {x^2/(1-x) - Sum_{k>=1} x^(2^k)/(1-x^(2^k))}. - Ralf 
               Stephan (ralf(AT)ark.in-berlin.de), Apr 13 2002
%F A048881 C(n, k) is the number of occurrence of k in the n-th group of terms in 
               this sequence read by rows: {0}; {0, 1}; {0, 1, 1, 2}; {0, 1, 1, 
               2, 1, 2, 2, 3}; {0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4 }; 
               ... - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 01 2004
%e A048881 Contribution from Omar E. Pol (info(AT)polprimos.com), Jun 07 2009: (Start)
%e A048881 Triangle begins:
%e A048881 0;
%e A048881 0,1;
%e A048881 0,1,1,2;
%e A048881 0,1,1,2,1,2,2,3;
%e A048881 0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4;
%e A048881 0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5;
%e A048881 (End)
%o A048881 (PARI) { a(n) = if( n<0, 0, n++; n /= 2^valuation(n,2); subst( Pol( binary( 
               n ) ), x, 1) - 1 ) } /* Michael Somos Aug 23 2007 */
%o A048881 (PARI) {a(n) = if( n<0, 0, valuation( (2*n)! / n! / (n+1)!, 2 ) ) } /
               * Michael Somos Aug 23 2007 */
%Y A048881 Cf. A000120, A006257, A007318, A063250.
%Y A048881 First differences of A078903.
%Y A048881 Cf. A000079, A001787. [From Omar E. Pol (info(AT)polprimos.com), Jun 
               07 2009]
%Y A048881 Sequence in context: A066360 A061358 A025866 this_sequence A026931 A127506 
               A007968
%Y A048881 Adjacent sequences: A048878 A048879 A048880 this_sequence A048882 A048883 
               A048884
%K A048881 nonn,tabf,new
%O A048881 0,7
%A A048881 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
%E A048881 Entry revised by N. J. A. Sloane, Jun 07 2009