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A120777 One half of denominators of partial sums of a series for sqrt(2). 11
1, 4, 8, 64, 128, 512, 1024, 16384, 32768, 131072, 262144, 2097152, 4194304, 16777216, 33554432, 1073741824, 2147483648, 8589934592, 17179869184, 137438953472, 274877906944, 1099511627776, 2199023255552 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.

FORMULA

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)

MAPLE

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

MATHEMATICA

Table[ Denominator[ CatalanNumber[k]/(-4)^k], {k, 0, 22}] (* Jean-François Alcover, Jun 21 2013 *)

CROSSREFS

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

KEYWORD

nonn,easy,frac

AUTHOR

Wolfdieter Lang, Jul 20 2006

STATUS

approved

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Last modified December 5 15:27 EST 2022. Contains 358588 sequences. (Running on oeis4.)