Search: id:A060818
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%I A060818
%S A060818 1,1,2,2,8,8,16,16,128,128,256,256,1024,1024,2048,2048,32768,32768,65536,
%T A060818 65536,262144,262144,524288,524288,4194304,4194304,8388608,8388608,
%U A060818 33554432,33554432,67108864,67108864,2147483648,2147483648,4294967296
%N A060818 2^(floor(n/2)+floor(n/4)+floor(n/8)+floor(n/16)+...).
%C A060818 a(n) is the size of the Sylow 2-subgroup of the symmetric group S_n.
%C A060818 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!.
%C A060818 Denominator of e(n,n) (see Maple line).
%C A060818 Denominator of the coefficient of x^n in n-th Legendre polynomial; numerators 
               are in A001790. - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 
               29 2002
%D A060818 V. H. Moll. The evaluation of integrals: a personal story, Notices Amer. 
               Math. Soc., 49 (No. 3, March 2002), 311-317.
%H A060818 Harry J. Smith, <a href="b060818.txt">Table of n, a(n) for n=0,...,200</
               a>
%H A060818 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RandomWalk1-Dimensional.html">Random Walk 1-Dimensional</a>
%H A060818 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LegendrePolynomial.html">Legendre Polynomial</a>
%F A060818 a(n) = 2^(A011371(n)) = 2^(floor(n/2)+floor(n/4)+floor(n/8)+floor(n/16)+...)
%F A060818 a(n)=gcd(n!, 2^n). - Labos E. (labos(AT)ana.sote.hu), Apr 22 2003
%F A060818 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).
%F A060818 L(n) = (2*n-1)!!/n!, with the double factorial (2*n-1)!!= A001147(n), 
               n>=0.
%e A060818 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, ...
%p A060818 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;
%t A060818 Table[GCD[w!, 2^w], {w, 1, 100}]
%o A060818 (PARI) a(n)=denominator(polcoeff(pollegendre(n),n))
%o A060818 (PARI) a(n)=if(n<0,0,2^sum(k=1,n,n\2^k))
%o A060818 (PARI) { for (n=0, 200, s=0; d=2; while (n>=d, s+=n\d; d*=2); write("b060818.txt", 
               n, " ", 2^s); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               Jul 12 2009]
%Y A060818 Cf. A011371, A001790. a(n)=A046161([n/2]).
%Y A060818 Row sums of triangle A100258.
%Y A060818 Cf. A100258.
%Y A060818 Sequence in context: A151924 A058524 A072576 this_sequence A082887 A137583 
               A099328
%Y A060818 Adjacent sequences: A060815 A060816 A060817 this_sequence A060819 A060820 
               A060821
%K A060818 nonn,frac
%O A060818 0,3
%A A060818 Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 29 2001
%E A060818 Additional comments from Henry Bottomley (se16(AT)btinternet.com), May 
               01 2001

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