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%I #76 Feb 28 2022 10:26:07
%S 1,5,11,93,193,793,1619,26333,53381,215955,436109,3518265,7088533,
%T 28539857,57414019,1846943453,3711565741,14911085359,29941580393,
%U 240416274739,482473579583,1936010885087,3883457090629,62306843256889
%N Numerators of partial sums of Catalan numbers scaled by powers of 1/4.
%C For denominators see A120777.
%C From the expansion of 0 = sqrt(1-1) = 1 - (1/2)*Sum_{k>=0} C(k)/4^k one has r:=lim_{n->infinity} r(n) = 2, with the partial sums r(n) defined below.
%C The series a(n)/A046161(n+1) is absolutely convergent to 1. - _Ralf Steiner_, Feb 16 2017
%C If n >= 1 it appears a(n-1) is equal to the difference between the denominator and the numerator of the ratio (2n)!!/(2n-1)!!. In particular a(n-1) is the difference between the denominator and the numerator of the ratio A001147(2n-1)/A000165(2n). See examples. - _Anthony Hernandez_, Feb 05 2020
%C From _Peter Bala_, Feb 16 2022: (Start)
%C Sum_{k = 0..n-1} Catalan(k)/4^k = (1/4^n)*(2*n)*binomial(2*n,n) *( 1 - 1/(1*2)*(n-1)/(n+1) - 1/(2*3)*(n-1)*(n-2)/((n+1)*(n+2)) - 1/(3*4)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) - 1/(4*5)*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) - ... ). Cf. A082687 and A101028.
%C This identity allows us to extend the definition of Sum_{k = 0..n} Catalan(k)/4^k to non-integral values of n. (End)
%H Vincenzo Librandi, <a href="/A120778/b120778.txt">Table of n, a(n) for n = 0..100</a>
%H Wolfdieter Lang, <a href="/A120778/a120778.txt">Rationals r(n) and limit 2</a>
%F a(n) = numerator(r(n)), with the rationals r(n):=Sum_{k = 0..n} C(k)/4^k with C(k) := A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.
%F r(n) = (4/Pi)*(n+1)*Integral_{x = 0..1} x^n*arcsin(sqrt(x)) dx. - _Groux Roland_, Jan 03 2011
%F r(n) = 2*(1 - binomial(2*n+2,n+1)/4^(n+1)). - _Groux Roland_, Jan 04 2011
%F a(n) = A141244(2n+2) = A141244(2n+3) (conjectural). - _Greg Martin_, Aug 16 2014, corrected by _M. F. Hasler_, Aug 18 2014
%F From _Peter Luschny_, Dec 21 2017: (Start)
%F a(n) = numerator(1 - ((n+1/2)!)/(sqrt(Pi)*(n+1)!)).
%F a(n) = 2^(2*(n+1) - HammingWeight(n+1))*(1 - ((n+1/2)!)/(sqrt(Pi)*(n+1)!)). (End)
%e Rationals r(n): [1, 5/4, 11/8, 93/64, 193/128, 793/512, 1619/1024, 26333/16384, ...].
%e From _Anthony Hernandez_, Feb 05 2020: (Start)
%e For n = 4. The 4th even number is 8, and 8!!/(8-1)!! = 128/35, so a(4-1) = a(3) = 128 - 35 = 93.
%e For n = 7. The 7th even number is 14, and 14!!/(14-1)!! = 2048/429, so a(7-1) = a(6) = 2048 - 429 = 1619. (End)
%p 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
%t 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 *)
%t a[n_] := 2^(2(n+1) - DigitCount[n+1,2,1])(1 - ((n+1/2)!)/(Sqrt[Pi](n+1)!));
%t Table[a[n], {n, 0, 23}] (* _Peter Luschny_, Dec 21 2017 *)
%o (Magma) [Numerator(2*(1-Binomial(2*n+2,n+1)/4^(n+1))): n in [0..25]]; // _Vincenzo Librandi_, Feb 17 2017
%Y Factors of A160481. Cf. A120777 (denominators), A082687, A101028, A141244.
%K nonn,easy,frac
%O 0,2
%A _Wolfdieter Lang_, Jul 20 2006
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