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A120778
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Numerators of partial sums of Catalan numbers scaled by powers of 1/4.
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5
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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; internal format)
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OFFSET
| 0,2
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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.
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LINKS
| W. Lang: Rationals r(n) and limit 2.
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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
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EXAMPLE
| Rationals r(n): [1, 5/4, 11/8, 93/64, 193/128, 793/512, 1619/1024, 26333/16384,...].
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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 *)
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CROSSREFS
| Factor of A160481 - Johannes W. Meijer, May 24 2009
Sequence in context: A057727 A128454 A188514 * A042761 A123025 A053778
Adjacent sequences: A120775 A120776 A120777 * A120779 A120780 A120781
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KEYWORD
| nonn,easy,frac
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 20 2006
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