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%I #120 Oct 31 2022 07:34:43
%S 1,1,2,1,2,2,4,1,2,2,4,2,4,4,8,1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,1,2,2,
%T 4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,1,2,
%U 2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,2,4,4
%N a(n) = 2^(A000120(n+1) - 1), n >= 0.
%C a(n) = 2^A048881 = 2^{maximal power of 2 dividing the n-th Catalan number (A000108)}. [Comment corrected by _N. J. A. Sloane_, Apr 30 2018]
%C Row sums of triangle A128937. - _Philippe Deléham_, May 02 2007
%C a(n) = sum of (n+1)-th row terms of triangle A167364. - _Gary W. Adamson_, Nov 01 2009
%C a(n), n >= 1: Numerators of Maclaurin series for 1 - ((sin x)/x)^2, A117972(n), n >= 2: Denominators of Maclaurin series for 1 - ((sin x)/x)^2, the correlation function in Montgomery's pair correlation conjecture. - _Daniel Forgues_, Oct 16 2011
%C For n > 0: a(n) = A007954(A007931(n)). - _Reinhard Zumkeller_, Oct 26 2012
%C a(n) = A261363(2*(n+1), n+1). - _Reinhard Zumkeller_, Aug 16 2015
%C From _Gus Wiseman_, Oct 30 2022: (Start)
%C Also the number of coarsenings of the (n+1)-th composition in standard order. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See link for sequences related to standard compositions. For example, the a(10) = 4 coarsenings of (2,1,1) are: (2,1,1), (2,2), (3,1), (4).
%C Also the number of times n+1 appears in A357134. For example, 11 appears at positions 11, 20, 33, and 1024, so a(10) = 4.
%C (End)
%H Reinhard Zumkeller, <a href="/A048896/b048896.txt">Table of n, a(n) for n = 0..10000</a>
%H Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, <a href="https://ojs.lib.uwo.ca/index.php/maple/article/view/14037">Some Facts and Conjectures about Mandelbrot Polynomials</a>, Maple Transactions (2021) Vol. 1, No. 1 Article 1.
%H Neil J. Calkin, Eunice Y. S. Chan, Robert M. Corless, David J. Jeffrey, and Piers W. Lawrence, <a href="https://arxiv.org/abs/2104.01116">A Fractal Eigenvector</a>, arXiv:2104.01116 [math.DS], 2021.
%H Emeric Deutsch and B. E. Sagan, <a href="https://arxiv.org/abs/math/0407326">Congruences for Catalan and Motzkin numbers and related sequences</a>, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.
%H OEIS Wiki, <a href="/wiki/Montgomery%27s_pair_correlation_conjecture">Montgomery's pair correlation conjecture</a>
%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%F a(n) = 2^A048881(n).
%F a(n) = 2^k if 2^k divides A000108(n) but 2^(k+1) does not divide A000108(n).
%F It appears that a(n) = Sum_{k=0..n} binomial(2*(n+1), k) mod 2. - Christopher Lenard (c.lenard(AT)bendigo.latrobe.edu.au), Aug 20 2001
%F a(0) = 1; a(2*n) = 2*a(2*n-1); a(2*n+1) = a(n).
%F a(n) = (1/2) * A001316(n+1). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004
%F It appears that a(n) = Sum_{k=0..2n} floor(binomial(2n+2, k+1)/2)(-1)^k = 2^n - Sum_{k=0..n+1} floor(binomial(n+1, k)/2). - _Paul Barry_, Dec 24 2004
%F a(n) = Sum_{k=0..n} (T(n,k) mod 2) where T = A039598, A053121, A052179, A124575, A126075, A126093. - _Philippe Deléham_, May 02 2007
%F a(n) = numerator(b(n)), where sin(x)^2/x = Sum_{n>0} b(n)*(-1)^n x^(2*n-1). - _Vladimir Kruchinin_, Feb 06 2013
%F a((2*n+1)*2^p-1) = A001316(n), p >= 0 and n >= 0. - _Johannes W. Meijer_, Feb 12 2013
%F a(n) = numerator(2^n / (n+1)!). - _Vincenzo Librandi_, Apr 12 2014
%F a(2n) = (2n+1)!/(n!n!)/A001803(n). - _Richard Turk_, Aug 23 2017
%F a(2n-1) = (2n-1)!/(n!(n-1)!)/A001790(n). - _Richard Turk_, Aug 23 2017
%e From _Omar E. Pol_, Jul 21 2009: (Start)
%e If written as a triangle:
%e 1;
%e 1,2;
%e 1,2,2,4;
%e 1,2,2,4,2,4,4,8;
%e 1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16;
%e 1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32;
%e ...,
%e the first half-rows converge to Gould's sequence A001316.
%e (End)
%p a := n -> 2^(add(i,i=convert(n+1,base,2))-1): seq(a(n), n=0..97); # _Peter Luschny_, May 01 2009
%t NestList[Flatten[#1 /. a_Integer -> {a, 2 a}] &, {1}, 4] // Flatten (* _Robert G. Wilson v_, Aug 01 2012 *)
%t Table[Numerator[2^n / (n + 1)!], {n, 0, 200}] (* _Vincenzo Librandi_, Apr 12 2014 *)
%t Denominator[Table[BernoulliB[2*n] / (Zeta[2*n]/Pi^[2*n]), {n, 1, 100}]] (* _Terry D. Grant_, May 29 2017 *)
%t Table[Denominator[((2 n)!/2^(2 n + 1)) (-1)^n], {n, 1, 100}]/4 (* _Terry D. Grant_, May 29 2017 *)
%t 2^IntegerExponent[CatalanNumber[Range[0,100]],2] (* _Harvey P. Dale_, Apr 30 2018 *)
%o (PARI) a(n)=if(n<1,1,if(n%2,a(n/2-1/2),2*a(n-1)))
%o (PARI) a(n) = 1 << (hammingweight(n+1)-1); \\ _Kevin Ryde_, Feb 19 2022
%o (Haskell)
%o a048896 n = a048896_list !! n
%o a048896_list = f [1] where f (x:xs) = x : f (xs ++ [x,2*x])
%o -- _Reinhard Zumkeller_, Mar 07 2011
%o (Haskell)
%o import Data.List (transpose)
%o a048896 = a000079 . a000120
%o a048896_list = 1 : concat (transpose
%o [zipWith (-) (map (* 2) a048896_list) a048896_list,
%o map (* 2) a048896_list])
%o -- _Reinhard Zumkeller_, Jun 16 2013
%o (Magma) [Numerator(2^n / Factorial(n+1)): n in [0..100]]; // _Vincenzo Librandi_, Apr 12 2014
%Y This is Guy Steele's sequence GS(3, 5) (see A135416).
%Y Equals first right hand column of triangle A160468.
%Y Equals A160469(n+1)/A002425(n+1).
%Y Cf. A000079, A001316, A117972, A160476, A167364, A220466, A261363.
%Y Standard compositions are listed by A066099.
%Y The opposite version (counting refinements) is A080100.
%Y The version for Heinz numbers of partitions is A317141.
%Y Cf. A000120, A001511, A029931, A048793, A058891, A070939, A272919, A357134.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_
%E New definition from _N. J. A. Sloane_, Mar 01 2008
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