

A120778


Numerators of partial sums of Catalan numbers scaled by powers of 1/4.


6



1, 5, 11, 93, 193, 793, 1619, 26333, 53381, 215955, 436109, 3518265, 7088533, 28539857, 57414019, 1846943453, 3711565741, 14911085359, 29941580393, 240416274739, 482473579583, 1936010885087, 3883457090629, 62306843256889
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

For denominators see A120777.
From the expansion of 0 = sqrt(11) = 1(1/2)*Sum(C(k)/4^k,k=0..infinity) one has r:=limit(r(n),n to infinity)=2, with the partial sums r(n) defined below.
The series a(n)/A046161(n+1) is absolutely convergent to 1.  Ralf Steiner, Feb 16 2017
If n >= 1 it appears a(n1) is equal to the difference between the denominator and the numerator of the ratio (2n)!!/(2n1)!!. In particular a(n1) is the difference between the denominator and the numerator of the ratio A001147(2n1)/A000165(2n). See examples.  Anthony Hernandez, Feb 05 2020


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100
W. Lang, Rationals r(n) and limit 2.


FORMULA

a(n) = numerator(r(n)), with the rationals r(n):=sum(C(k)/4^k,k=0..n) with C(k):=A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.
r(n) = (4/Pi)*(n+1)*int(x^n*arcsin(sqrt(x)),x=0..1).  Roland Groux, Jan 03 2011
r(n) = 2*[1binomial(2*n+2,n+1)/4^(n+1)].  Roland Groux, Jan 04 2011
a(n) = A141244(2n+2) = A141244(2n+3) (conjectural).  Greg Martin, Aug 16 2014, corrected by M. F. Hasler, Aug 18 2014
From Peter Luschny, Dec 21 2017: (Start)
a(n) = numerator(1((n+1/2)!)/(sqrt(Pi)*(n+1)!)).
a(n) = 2^(2*(n+1)  HammingWeight(n+1))*(1  ((n+1/2)!)/(sqrt(Pi)*(n+1)!)). (End)


EXAMPLE

Rationals r(n): [1, 5/4, 11/8, 93/64, 193/128, 793/512, 1619/1024, 26333/16384, ...].
From Anthony Hernandez, Feb 05 2020: (Start)
For n=4. The 4th even number is 8, and 8!!/(81)!! = 128/35, so a(41) = a(3) = 128  35 = 93.
For n=7. The 7th even number is 14, and 14!!/(141)!! = 2048/429, so a(71) = a(6) = 2048  429 = 1619. (End)


MAPLE

a := n > 2^(2*(n+1)  add(i, i=convert(n+1, base, 2)))* (1((n+1/2)!)/(sqrt(Pi)*(n+1)!)): seq(simplify(a(n)), n=0..23); # Peter Luschny, Dec 21 2017


MATHEMATICA

f[n_] := f[n] = Numerator[(4/Pi) (n + 1) Integrate[x^n*ArcSin[Sqrt[x]], {x, 0, 1}]]; Array[f, 23, 0] (* Robert G. Wilson v, Jan 03 2011 *)
a[n_] := 2^(2(n+1)  DigitCount[n+1, 2, 1])(1  ((n+1/2)!)/(Sqrt[Pi](n+1)!));
Table[a[n], {n, 0, 23}] (* Peter Luschny, Dec 21 2017 *)


PROG

(MAGMA) [Numerator(2*(1Binomial(2*n+2, n+1)/4^(n+1))): n in [0..25]]; // Vincenzo Librandi, Feb 17 2017


CROSSREFS

Factor of A160481.  Johannes W. Meijer, May 24 2009
Cf. A141244.  Greg Martin, Aug 16 2014
Cf. A120777 (denominators).
Sequence in context: A128454 A188514 A301923 * A042761 A224270 A123025
Adjacent sequences: A120775 A120776 A120777 * A120779 A120780 A120781


KEYWORD

nonn,easy,frac


AUTHOR

Wolfdieter Lang, Jul 20 2006


STATUS

approved



