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A060818
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2^(floor(n/2)+floor(n/4)+floor(n/8)+floor(n/16)+...).
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10
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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; internal format)
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OFFSET
| 0,3
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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 (benoit7848c(AT)orange.fr), Nov 29 2002
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REFERENCES
| V. H. Moll. The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
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LINKS
| Harry J. Smith, Table of n, a(n) for n=0,...,200
Eric Weisstein's World of Mathematics, Random Walk 1-Dimensional
Eric Weisstein's World of Mathematics, Legendre Polynomial
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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 E. (labos(AT)ana.sote.hu), 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.
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EXAMPLE
| 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, ...
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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;
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MATHEMATICA
| Table[GCD[w!, 2^w], {w, 1, 100}]
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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); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 12 2009]
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CROSSREFS
| Cf. A011371, A001790. a(n)=A046161([n/2]).
Row sums of triangle A100258.
Cf. A100258.
Sequence in context: A151924 A058524 A072576 * A082887 A137583 A099328
Adjacent sequences: A060815 A060816 A060817 * A060819 A060820 A060821
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KEYWORD
| nonn,frac
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AUTHOR
| Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 29 2001
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EXTENSIONS
| Additional comments from Henry Bottomley (se16(AT)btinternet.com), May 01 2001
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