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A060818
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a(n) = 2^(n - HammingWeight(n)) = 2^(n - BitCount(n)) = 2^(n - A000120(n)).
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31
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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;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = 2^(floor(n/2) + floor(n/4) + floor(n/8) + floor(n/16) + ...).
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) = (n! XOR floor(n!/2)) XOR (n!-1 XOR floor((n!-1)/2)). - Gary Detlefs, Jun 13 2014
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EXAMPLE
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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, ... .
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MAPLE
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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;
HammingWeight := n -> add(convert(n, base, 2)):
seq(2^(n - HammingWeight(n)), n = 0..34); # Peter Luschny, Mar 23 2024
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MATHEMATICA
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Table[GCD[w!, 2^w], {w, 100}]
(* Second program, more efficient *)
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PROG
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(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)
(Magma) [1] cat [Denominator(Catalan(n)/2^n): n in [0..50]]; // Vincenzo Librandi, Sep 01 2014
(Python 3.10+)
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CROSSREFS
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KEYWORD
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nonn,easy,frac
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AUTHOR
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Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 29 2001
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EXTENSIONS
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STATUS
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approved
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