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

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• frac

Comments

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.

Links

• 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.

Crossrefs

Appears in A162446.

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

Programs

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 *)
(* Alternative: *)
A120777[n_] := 2^(2*n - IntegerExponent[CatalanNumber[n], 2]);
Table[A120777[n], {n, 0, 26}]  (* Peter Luschny, Apr 16 2024 *)

Formula

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

Extensions

New name by Peter Luschny, Apr 16 2024