A120778 Numerators of partial sums of Catalan numbers scaled by powers of 1/4.

1, 5, 11, 93, 193, 793, 1619, 26333, 53381, 215955, 436109, 3518265, 7088533, 28539857, 57414019, 1846943453, 3711565741, 14911085359, 29941580393, 240416274739, 482473579583, 1936010885087, 3883457090629, 62306843256889

Offset: 0

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• frac

Comments

For denominators see A120777.

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.

The series a(n)/A046161(n+1) is absolutely convergent to 1. - Ralf Steiner, Feb 16 2017

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

From Peter Bala, Feb 16 2022: (Start)

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.

This identity allows us to extend the definition of Sum_{k = 0..n} Catalan(k)/4^k to non-integral values of n. (End)

Examples

			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!!/(8-1)!! = 128/35, so a(4-1) = a(3) = 128 - 35 = 93.
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)
		

Links

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

Crossrefs

Factors of A160481. Cf. A120777 (denominators), A082687, A101028, A141244.

Programs

Magma

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

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 *)

Formula

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.

r(n) = (4/Pi)*(n+1)*Integral_{x = 0..1} x^n*arcsin(sqrt(x)) dx. - Groux Roland, Jan 03 2011

r(n) = 2*(1 - binomial(2*n+2,n+1)/4^(n+1)). - Groux Roland, 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)