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A048896 a(n) = 2^(A000120(n+1) - 1), n >= 0. 33
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

Row sums of triangle A128937. - Philippe Deléham, May 02 2007

a(n) = sum of (n+1)-th row terms of triangle A167364. - Gary W. Adamson, Nov 01 2009

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

For n > 0: a(n) = A007954(A007931(n)). - Reinhard Zumkeller, Oct 26 2012

a(n) = A261363(2*(n+1), n+1). - Reinhard Zumkeller, Aug 16 2015

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

OEIS Wiki, Montgomery's pair correlation conjecture

FORMULA

a(n) = 2^A048881(n).

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

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

a(0) = 1; a(2*n) = 2*a(2*n-1); a(2*n+1) = a(n).

a(n) = (1/2) * A001316(n+1). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004

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

a(n) = Sum_{k=0..n} (T(n,k) mod 2) where T = A039598, A053121, A052179, A124575, A126075, A126093. - Philippe Deléham, May 02 2007

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

a((2*n+1)*2^p-1) = A001316(n), p >= 0 and n >= 0. - Johannes W. Meijer, Feb 12 2013

a(n) = numerator(2^n / (n+1)!). - Vincenzo Librandi, Apr 12 2014

a(2n) = (2n+1)!/(n!n!)/A001803(n). - Richard Turk, Aug 23 2017

a(2n-1) = (2n-1)!/(n!(n-1)!)/A001790(n). - Richard Turk, Aug 23 2017

EXAMPLE

From Omar E. Pol, Jul 21 2009: (Start)

If written as a triangle:

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;

1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32;

...,

the first half-rows converge to Gould's sequence A001316.

(End)

MAPLE

a := n -> 2^(add(i, i=convert(n+1, base, 2))-1): seq(a(n), n=0..97); # Peter Luschny, May 01 2009

MATHEMATICA

NestList[Flatten[#1 /. a_Integer -> {a, 2 a}] &, {1}, 4] // Flatten (* Robert G. Wilson v, Aug 01 2012 *)

Table[Numerator[2^n / (n + 1)!], {n, 0, 200}] (* Vincenzo Librandi, Apr 12 2014 *)

Denominator[Table[BernoulliB[2*n] / (Zeta[2*n]/Pi^[2*n]), {n, 1, 100}]] (* Terry D. Grant, May 29 2017 *)

Table[Denominator[((2 n)!/2^(2 n + 1)) (-1)^n], {n, 1, 100}]/4 (* Terry D. Grant, May 29 2017 *)

PROG

(PARI) a(n)=if(n<1, 1, if(n%2, a(n/2-1/2), 2*a(n-1)))

(Haskell)

a048896 n = a048896_list !! n

a048896_list = f [1] where f (x:xs) = x : f (xs ++ [x, 2*x])

-- Reinhard Zumkeller, Mar 07 2011

(Haskell)

import Data.List (transpose)

a048896 = a000079 . a000120

a048896_list = 1 : concat (transpose

   [zipWith (-) (map (* 2) a048896_list) a048896_list,

    map (* 2) a048896_list])

-- Reinhard Zumkeller, Jun 16 2013

(MAGMA) [Numerator(2^n / Factorial(n+1)): n in [0..100]]; // Vincenzo Librandi, Apr 12 2014

CROSSREFS

This is Guy Steele's sequence GS(3, 5) (see A135416).

Equals first right hand column of triangle A160468.

Equals A160469(n+1)/A002425(n+1).

Cf. A160476, A000079, A001316, A167364, A220466, A001316, A080100, A261363, A117972.

Sequence in context: A238212 A255723 A214718 * A130831 A151678 A273126

Adjacent sequences:  A048893 A048894 A048895 * A048897 A048898 A048899

KEYWORD

nonn

AUTHOR

Wolfdieter Lang

EXTENSIONS

New definition from N. J. A. Sloane, Mar 01 2008

STATUS

approved

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Last modified February 23 00:33 EST 2018. Contains 299473 sequences. (Running on oeis4.)