0,7

Highest power of 2 dividing n-th Catalan number (A000108).

a(n) = 0 iff n = 2^k - 1, k=0,1,...

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.

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.

Highest power of 2 dividing binomial(n,floor(n/2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 20 2003

2^a(n) are numerators in the maclaurin series for (sin x)^2. [From Jacob A. Siehler (siehlerj(AT)wlu.edu), Nov 11 2009]

R. Alter and K. K. Kubota, Prime and Prime Power Divisibility of Catalan Numbers, J. Com. Th. A 15 (1973) pp. 243-256.

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

a(n)= k if 2^k divides A000108(n) but 2^(k+1) does not divide A000108(n).

a(2*n) = a(n-1)+1, a(2*n+1) = a(n). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 10 2002

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

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

Contribution from Omar E. Pol (info(AT)polprimos.com), Jun 07 2009: (Start)

Triangle begins:

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;

0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5;

(End)

(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 */

(PARI) {a(n) = if( n<0, 0, valuation( (2*n)! / n! / (n+1)!, 2 ) ) } /* Michael Somos Aug 23 2007 */

Cf. A000120, A006257, A007318, A063250.

First differences of A078903.

Cf. A000079, A001787. [From Omar E. Pol (info(AT)polprimos.com), Jun 07 2009]

Sequence in context: A066360 A061358 A025866 this_sequence A026931 A127506 A007968

Adjacent sequences: A048878 A048879 A048880 this_sequence A048882 A048883 A048884

nonn,tabf,new

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

Entry revised by N. J. A. Sloane, Jun 07 2009