A061549 Denominator of probability that there is no error when average of n numbers is computed, assuming errors of +1, -1 are possible and they each occur with p=1/4.

1, 8, 128, 1024, 32768, 262144, 4194304, 33554432, 2147483648, 17179869184, 274877906944, 2199023255552, 70368744177664, 562949953421312, 9007199254740992, 72057594037927936, 9223372036854775808, 73786976294838206464, 1180591620717411303424, 9444732965739290427392

Offset: 0

Author

Leah Schmelzer (leah2002(AT)mit.edu), May 16 2001

Keywords

• nonn
• frac
• easy

Comments

We observe that b(n) = log(a(n))/log(2) = A120738(n). Furthermore c(n+1) = b(n+1)-b(n) = A090739(n+1) and c(n+1)-3 = A007814(n+1) for n>=0. - Johannes W. Meijer, Jul 06 2009

Using WolframAlpha, it appears that 2*a(n) gives the coefficients of Pi in the denominators of the residues of f(z) = 2z choose z at odd negative half integers. E.g., the residues of f(z) at z = -1/2, -3/2, -5/2 are 1/(2*Pi), 1/(16*Pi), and 3/(256*Pi) respectively. - Nicholas Juricic, Mar 31 2022

Examples

			For n=1, the binomial(2*n-1/2, -1/2) yields the term 3/8. The denominator of this term is 8, which is the second term of the sequence.
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..500
• Robert M. Kozelka, Grade Point Averages and the Central Limit Theorem, American Mathematical Monthly. Nov. 1979 (86:9) pp. 773-7.
• Eric Weisstein's World of Mathematics, Circle Line Picking
• Eric Weisstein's World of Mathematics, Gamma Function

Crossrefs

Cf. A007814, A061548, A090739.

Bisection of A046161.

Appears in A162448.

Programs

Magma

A061549:= func< n | 2^(4*n-(&+Intseq(2*n, 2))) >;
[A061549(n): n in [0..30]]; // G. C. Greubel, Oct 20 2024

Maple

seq(denom(binomial(2*n-1/2, -1/2)), n=0..20);

Mathematica

Table[Denominator[(4*n)!/(2^(4*n)*(2*n)!^2) ], {n, 0, 20}] (* Indranil Ghosh, Mar 11 2017 *)

PARI

for(n=0, 20, print1(denominator((4*n)!/(2^(4*n)*(2*n)!^2)),", ")) \\ Indranil Ghosh, Mar 11 2017

Python

import math
f = math.factorial
def A061549(n): return (2**(4*n)*f(2*n)**2) // math.gcd(f(4*n), (2**(4*n)*f(2*n)**2)) # Indranil Ghosh, Mar 11 2017

Sage

# uses[A000120]
def a(n): return 1 << (4*n - A000120(n))
[a(n) for n in (0..19)]  # Peter Luschny, Dec 02 2012

Formula

a(n) = denominator of binomial(2*n-1/2, -1/2).

a(n) are denominators of coefficients of 1/(sqrt(1+x)-sqrt(1-x)) power series. - Benoit Cloitre, Mar 12 2002

a(n) = 16^n/A001316(n). - Paul Barry, Jun 29 2006

a(n) = denom((4*n)!/(2^(4*n)*(2*n)!^2)). - Johannes W. Meijer, Jul 06 2009

a(n) = abs(A067624(n)/A117972(n)). - Johannes W. Meijer, Jul 06 2009

Extensions

More terms from Asher Auel, May 20 2001