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A120777
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One half of denominators of partial sums of a series for sqrt(2).
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10
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1, 4, 8, 64, 128, 512, 1024, 16384, 32768, 131072, 262144, 2097152, 4194304, 16777216, 33554432, 1073741824, 2147483648, 8589934592, 17179869184, 137438953472, 274877906944, 1099511627776, 2199023255552
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OFFSET
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0,2
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COMMENTS
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Also denominators of partial sums sum(C(k)/(-4)^k, k=0..n) = A120788(n)/A120777(n).
One half of denominators of partial sums which involve Catalan numbers A000108(k) divided by 4^k with alternating signs.
The listed numbers coincide with the denominators of sum(C(k)/4^k, k=0..n). See numerators A120778. In general these denominators may be different. See e.g. A120783 versus A120793 and A120787 versus A120796.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
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a(n) = denominator(r(n)), with the rationals r(n) defined under A120088.
Contribution from Johannes W. Meijer, Jul 06 2009: (Start)
a(n) = denom(C(2*n+2,n+1)/2^(2*n+1)).
If b(n) = log(a(n))/log(2) then c(n) = b(n+1)-b(n) = A001511(n+1) i.e. the ruler function.
(End)
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MAPLE
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a := n -> denom(binomial(2*n+2, n+1) / 2^(2*n+1)):
seq(a(n), n=0..22); # Johannes W. Meijer, Sep 23 2012
Conjecture: The following Maple program appears to generate this sequence! Z[0]:=0: for k to 30 do Z[k]:=simplify(1/(2-z*Z[k-1])) od: g:=sum((Z[j]-Z[j-1]), j=1..30): gser:=series(g, z=0, 27): seq(denom(coeff(gser, z, n))/2, n=0..22); # Zerinvary Lajos, May 21 2008
a := proc(n) option remember: if n = 0 then b(0):=0 else b(n) := b(n-1) + A001511(n+1) fi: a(n) := 2^b(n) end proc: A001511 := proc(n) option remember: if n = 1 then 1 else procname(n-1) + (-1)^n * procname(floor(n/2)) fi: end proc:
seq(a(n), n=0..22); # Johannes W. Meijer, Jul 06 2009, revised Sep 23 2012
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MATHEMATICA
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Table[ Denominator[ CatalanNumber[k]/(-4)^k], {k, 0, 22}] (* Jean-François Alcover, Jun 21 2013 *)
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CROSSREFS
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Appears in A162446. [Johannes W. Meijer, Jul 06 2009]
Sequence in context: A275574 A214590 A215713 * A091095 A075787 A086891
Adjacent sequences: A120774 A120775 A120776 * A120778 A120779 A120780
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KEYWORD
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nonn,easy,frac
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AUTHOR
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Wolfdieter Lang, Jul 20 2006
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STATUS
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approved
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