A120788 -id:A120788 - cp's OEIS frontend

A120777 a(n) = 2^(2*n - valuation(CatalanNumber(n), 2)).

1, 4, 8, 64, 128, 512, 1024, 16384, 32768, 131072, 262144, 2097152, 4194304, 16777216, 33554432, 1073741824, 2147483648, 8589934592, 17179869184, 137438953472, 274877906944, 1099511627776, 2199023255552, 35184372088832, 70368744177664, 281474976710656, 562949953421312

Offset: 0

Wolfdieter Lang, Jul 20 2006

Previous name: One half of denominators of partial sums of a series for sqrt(2).
Also denominators of partial sums Sum_{k=0..n} (C(k)/(-4)^k) = 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.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.

Appears in A162446.

Cf. A048881, A120777, A120783, A120787, A120788, A120793, A120796, A283208.

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

Table[Denominator[CatalanNumber[k]/(-4)^k], {k, 0, 22}] (* Jean-François Alcover, Jun 21 2013 *)
(* Alternative: *)
A120777[n_] := 2^(2*n - IntegerExponent[CatalanNumber[n], 2]);
Table[A120777[n], {n, 0, 26}]  (* Peter Luschny, Apr 16 2024 *)

a(n) = denominator(r(n)), with the rationals r(n) defined under A120088.
From Johannes W. Meijer, Jul 06 2009: (Start)
a(n) = denominator(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)
a(n) = 2^(2*n- A048881(n)) = 2^A283208(n). - Amiram Eldar, Apr 18 2024

New name by Peter Luschny, Apr 16 2024

A120088 Numerators of partial sums of a series for sqrt(2).

Original entry on oeis.org

3, 11, 23, 179, 365, 1439, 2911, 46147, 93009, 369605, 743409, 5917879, 11887761, 47365319, 95064943, 3032383331, 6082445497, 24264959593, 48649328861, 388310999293, 778263028691, 3106935548009, 6225306416473, 99433372856743, 199189221750317, 795541400400905

Offset: 0

Wolfdieter Lang, Jul 20 2006

Involving alternating sums over scaled Catalan numbers, A000108(k)/4^k.
From the expansion of sqrt(1+x) = 1 + x*(Sum_{k>=0} C(k)*(-x/4)^k)/2, valid for |x|<=1, one finds for x=+1: sqrt(2) = 1 + (Sum_{k>=0} (-1)^k*C(k)/4^k)/2.
The denominators are given by 2*A120777(n).
The rationals r(n):=1 + (Sum_{k=0..n} (-1)^k*C(k)/4^k)/2, with the Catalan numbers C(n)=A000108(n), are A120088(n)/(2*A120777(n)), n>=0.

			Rationals r(n): [3/2, 11/8, 23/16, 179/128, 365/256, 1439/1024, 2911/2048, 46147/32768,...]

G. C. Greubel, Table of n, a(n) for n = 0..1000
W. Lang, Rationals r(n).

For similar partial sums with positive terms (not alternating) see rationals A119951/A120069.

For the partial sums (Sum_{k=0..n} (-1)^k*C(k)/4^k) see A120788(n)/A120777(n).

[Numerator(1 + (&+[(-1/4)^k*Binomial(2*k,k)/(2*(k+1)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, Mar 27 2018

r[n_]:= 1+Sum[(-1/4)^k*CatalanNumber[k]/2, {k, 0, n}]; Numerator[Table[ r[n], {n, 0, 50}]] (* G. C. Greubel, Mar 27 2018 *)

{r(n) = 1 + sum(k=0,n, (-1/4)^k*binomial(2*k,k)/(2*(k+1)))};
for(n=0,30, print1(numerator(r(n)), ", ")) \\ G. C. Greubel, Mar 27 2018

a(n) = numerator(r(n)), with the rationals defined above.

cp's OEIS Frontend

A120777 a(n) = 2^(2*n - valuation(CatalanNumber(n), 2)).

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