|
| |
|
|
A120777
|
|
One half of denominators of partial sums of a series for sqrt(2).
|
|
10
|
|
|
|
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
|
| |
|
|
