A060818 a(n) = 2^(n - HammingWeight(n)) = 2^(n - BitCount(n)) = 2^(n - A000120(n)).

1, 1, 2, 2, 8, 8, 16, 16, 128, 128, 256, 256, 1024, 1024, 2048, 2048, 32768, 32768, 65536, 65536, 262144, 262144, 524288, 524288, 4194304, 4194304, 8388608, 8388608, 33554432, 33554432, 67108864, 67108864, 2147483648, 2147483648, 4294967296

Offset: 0

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 29 2001

Keywords

• nonn
• easy
• frac

Comments

a(n) is the size of the Sylow 2-subgroup of the symmetric group S_n.

Also largest power of 2 which is a factor of n! and (apart from a(3)) the largest perfect power which is a factor of n!.

Denominator of e(n,n) (see Maple line).

Denominator of the coefficient of x^n in n-th Legendre polynomial; numerators are in A001790. - Benoit Cloitre, Nov 29 2002

Examples

			G.f. = 1 + x + 2*x^2 + 2*x^3 + 8*x^4 + 8*x^5 + 16*x^6 + 16*x^7 + 128*x^8 + ...
e(n,n) sequence begins 1, 1, 3/2, 5/2, 35/8, 63/8, 231/16, 429/16, 6435/128, 12155/128, 46189/256, ... .
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, 87(2) (2014), 135-143.
• V. H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49(3) (2002), 311-317.
• Eric Weisstein's World of Mathematics, Random Walk 1-Dimensional.
• Eric Weisstein's World of Mathematics, Legendre Polynomial.
• Index to divisibility sequences

Crossrefs

Cf. A000120, A005187, A001790, A011371, A334907.

a(n) = A046161([n/2]).

Row sums of triangle A100258.

Programs

Magma

[1] cat [Denominator(Catalan(n)/2^n): n in [0..50]]; // Vincenzo Librandi, Sep 01 2014
(Python 3.10+)
def A060818(n): return 1<Chai Wah Wu, Jul 11 2022

Maple

e := proc(l,m) local k; add(2^(k-2*m) * binomial(2*m-2*k,m-k) * binomial(m+k,m) * binomial(k,l), k=l..m); end;
A060818 := proc(n) option remember; `if`(n=0,1,2^(padic[ordp](n,2))*A060818(n-1)) end: seq(A060818(i), i=0..34); # Peter Luschny, Nov 16 2012
HammingWeight := n -> add(convert(n, base, 2)):
seq(2^(n - HammingWeight(n)), n = 0..34); # Peter Luschny, Mar 23 2024

Mathematica

Table[GCD[w!, 2^w], {w, 100}]
(* Second program, more efficient *)
Array[2^(# - DigitCount[#, 2, 1]) &, 35, 0] (* Michael De Vlieger, Mar 23 2024 *)

PARI

{a(n) = denominator( polcoeff( pollegendre(n), n))};

PARI

{a(n) = if( n<0, 0, 2^sum(k=1, n, n\2^k))};

PARI

{ for (n=0, 200, s=0; d=2; while (n>=d, s+=n\d; d*=2); write("b060818.txt", n, " ", 2^s); ) } \\ Harry J. Smith, Jul 12 2009

Sage

def A060818(n):
    A005187 = lambda n: A005187(n//2) + n if n > 0 else 0
    return 2^A005187(n//2)
[A060818(i) for i in (0..34)]  # Peter Luschny, Nov 16 2012

Formula

a(n) = 2^(floor(n/2) + floor(n/4) + floor(n/8) + floor(n/16) + ...).

a(n) = 2^(A011371(n)).

a(n) = gcd(n!, 2^n). - Labos Elemer, Apr 22 2003

a(n) = denominator(L(n)) with rational L(n):=binomial(2*n,n)/2^n. L(n) is the leading coefficient of the Legendre polynomial P_n(x).

L(n) = (2*n-1)!!/n!, with the double factorial (2*n-1)!! = A001147(n), n>=0.

a(n) = Product_{i=1..n} A006519(i). - Tom Edgar, Apr 30 2014

a(n) = (n! XOR floor(n!/2)) XOR (n!-1 XOR floor((n!-1)/2)). - Gary Detlefs, Jun 13 2014

a(n) = denominator(Catalan(n-1)/2^(n-1)) for n>0. - Vincenzo Librandi, Sep 01 2014

a(2*n) = a(2*n+1) = 2^n*a(n). - Robert Israel, Sep 01 2014

a(n) = n!*A063079(n+1)/A334907(n). - Petros Hadjicostas, May 16 2020

Extensions

Additional comments from Henry Bottomley, May 01 2001

New name from Peter Luschny, Mar 23 2024