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%I #29 Apr 18 2024 04:22:04
%S 1,4,8,64,128,512,1024,16384,32768,131072,262144,2097152,4194304,
%T 16777216,33554432,1073741824,2147483648,8589934592,17179869184,
%U 137438953472,274877906944,1099511627776,2199023255552,35184372088832,70368744177664,281474976710656,562949953421312
%N a(n) = 2^(2*n - valuation(CatalanNumber(n), 2)).
%C Previous name: One half of denominators of partial sums of a series for sqrt(2).
%C Also denominators of partial sums Sum_{k=0..n} (C(k)/(-4)^k) = A120788(n)/A120777(n).
%C One half of denominators of partial sums which involve Catalan numbers A000108(k) divided by 4^k with alternating signs.
%C 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.
%H Vincenzo Librandi, <a href="/A120777/b120777.txt">Table of n, a(n) for n = 0..200</a>
%H Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, <a href="https://www.emis.de/journals/JIS/VOL21/Falcao/falcao2.html">Combinatorial Identities Associated with a Multidimensional Polynomial Sequence</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
%F a(n) = denominator(r(n)), with the rationals r(n) defined under A120088.
%F From _Johannes W. Meijer_, Jul 06 2009: (Start)
%F a(n) = denominator(C(2*n+2,n+1)/2^(2*n+1)).
%F 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)
%F a(n) = 2^(2*n- A048881(n)) = 2^A283208(n). - _Amiram Eldar_, Apr 18 2024
%p a := n -> denom(binomial(2*n+2, n+1) / 2^(2*n+1)):
%p seq(a(n), n=0..22); # _Johannes W. Meijer_, Sep 23 2012
%p 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
%p 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:
%p seq(a(n), n=0..22); # _Johannes W. Meijer_, Jul 06 2009, revised Sep 23 2012
%t Table[Denominator[CatalanNumber[k]/(-4)^k], {k, 0, 22}] (* _Jean-François Alcover_, Jun 21 2013 *)
%t (* Alternative: *)
%t A120777[n_] := 2^(2*n - IntegerExponent[CatalanNumber[n], 2]);
%t Table[A120777[n], {n, 0, 26}] (* _Peter Luschny_, Apr 16 2024 *)
%Y Appears in A162446.
%Y Cf. A048881, A120777, A120783, A120787, A120788, A120793, A120796, A283208.
%K nonn,easy,frac,changed
%O 0,2
%A _Wolfdieter Lang_, Jul 20 2006
%E New name by _Peter Luschny_, Apr 16 2024
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