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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

For denominators see A120777.

From the expansion of 0 = sqrt(1-1) = 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

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*[1-binomial(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

EXAMPLE

Rationals r(n): [1, 5/4, 11/8, 93/64, 193/128, 793/512, 1619/1024, 26333/16384, ...].

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 *)

PROG

(MAGMA) [Numerator(2*(1-Binomial(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: A228503 A128454 A188514 * A042761 A224270 A123025

Adjacent sequences:  A120775 A120776 A120777 * A120779 A120780 A120781

KEYWORD

nonn,easy,frac

AUTHOR

Wolfdieter Lang, Jul 20 2006

STATUS

approved

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Last modified November 18 19:06 EST 2017. Contains 294894 sequences.