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A060818 a(n) = 2^(floor(n/2) + floor(n/4) + floor(n/8) + floor(n/16) + ...). 21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

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, Vol. 87, No. 2, April 2014, pp. 135-143.

V. H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 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

FORMULA

a(n) = 2^(A011371(n)) = 2^(floor(n/2) + floor(n/4) + floor(n/8) + floor(n/16) + ...).

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

EXAMPLE

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, ... .

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

MATHEMATICA

Table[GCD[w!, 2^w], {w, 100}]

PROG

(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

(MAGMA) [1] cat [Denominator(Catalan(n)/2^n): n in [0..50]]; // Vincenzo Librandi, Sep 01 2014

CROSSREFS

Cf. A011371, A001790. a(n) = A046161([n/2]).

Row sums of triangle A100258.

Cf. A100258.

Sequence in context: A058524 A072576 A271342 * A082887 A137583 A099328

Adjacent sequences:  A060815 A060816 A060817 * A060819 A060820 A060821

KEYWORD

nonn,frac

AUTHOR

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

EXTENSIONS

Additional comments from Henry Bottomley, May 01 2001

STATUS

approved

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Last modified February 23 17:40 EST 2018. Contains 299584 sequences. (Running on oeis4.)