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
Examples
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.
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.
Programs
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Magma
A061548:= func< n | Numerator(Binomial(4*n,2*n)/4^n) >; [A061548(n): n in [0..25]]; // G. C. Greubel, Oct 19 2024
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Maple
seq(numer(binomial(2*n-1/2, -1/2)), n=0..20);
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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 *) -
PARI
for(n=0, 20, print1(numerator((4*n)!/(2^(4*n)*(2*n)!^2)),", ")) \\ Indranil Ghosh, Mar 11 2017
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Python
import math f = math.factorial def A061548(n): return f(4*n) // math.gcd(f(4*n), (2**(4*n)*f(2*n)**2)) # Indranil Ghosh, Mar 11 2017
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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
Formula
a(n) = numerator(binomial(2*n-1/2, -1/2)).
From Johannes W. Meijer, Jul 06 2009: (Start)
a(n) = numerator((4*n)!/(2^(4*n)*(2*n)!^2)).
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
Extensions
More terms from Asher Auel, May 20 2001