A061548 Numerator 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, 3, 35, 231, 6435, 46189, 676039, 5014575, 300540195, 2268783825, 34461632205, 263012370465, 8061900920775, 61989816618513, 956086325095055, 7391536347803839, 916312070471295267, 7113260368810144185, 110628135069209194801, 861577581086657669325, 26876802183334044115405

Offset

0,2

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.

Formula

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

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

a(n) = A001448(n)/A001316(n). - Peter Luschny, Mar 23 2014

a(n) is the numerator of the coefficient of power series in x around x=0 of sqrt(1 + sqrt(1 - x))/(sqrt(2)*sqrt(1 - x)). - Karol A. Penson, Apr 16 2018

Example

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

Maple

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

Mathematica

Table[Numerator[(4*n) !/(2^(4*n)*(2*n) !^2) ], {n, 0, 20}] (* Indranil Ghosh, Mar 11 2017 *)
Table[Numerator[SeriesCoefficient[Series[(Sqrt[1 + Sqrt[1 - x]]/Sqrt[2 - 2* x]), {x, 0, n}], n]], {n, 0, 20}]. (* Karol A. Penson, Apr 16 2018 *)

Prog

(Sage)
def A061548(n): return binomial(4*n, 2*n)/2^sum(n.digits(2))
[A061548(n) for n in (0..20)]  # Peter Luschny, Mar 23 2014
(PARI) for(n=0, 20, print1(numerator((4*n)!/(2^(4*n)*(2*n)!^2)), ", ")) \\ Indranil Ghosh, Mar 11 2017
(Python)
import math
from fractions import gcd
f = math.factorial
def A061548(n): return f(4*n) / gcd(f(4*n), (2**(4*n)*f(2*n)**2)) # Indranil Ghosh, Mar 11 2017

Crossrefs

Cf. A061549. Bisection of A001790.

Equals 2*A001448(n)/ A117973(n). - Johannes W. Meijer, Jul 06 2009

Sequence in context: A130061 A368391 A274875 * A019273 A271209 A202883

Adjacent sequences: A061545 A061546 A061547 * A061549 A061550 A061551

Keyword

nonn,frac,easy

Author

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

Extensions

More terms from Asher Auel, May 20 2001

Status

approved