cp's OEIS Frontend

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.

%I A061548 #46 Jul 27 2025 18:54:32
%S A061548 1,3,35,231,6435,46189,676039,5014575,300540195,2268783825,
%T A061548 34461632205,263012370465,8061900920775,61989816618513,
%U A061548 956086325095055,7391536347803839,916312070471295267,7113260368810144185,110628135069209194801,861577581086657669325,26876802183334044115405
%N 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.
%H A061548 Indranil Ghosh, <a href="/A061548/b061548.txt">Table of n, a(n) for n = 0..500</a>
%H A061548 Robert M. Kozelka, <a href="http://www.jstor.org/stable/2322033">Grade Point Averages and the Central Limit Theorem</a>, American Mathematical Monthly. Nov. 1979 (86:9) pp. 773-7.
%F A061548 a(n) = numerator(binomial(2*n-1/2, -1/2)).
%F A061548 From _Johannes W. Meijer_, Jul 06 2009: (Start)
%F A061548 a(n) = numerator((4*n)!/(2^(4*n)*(2*n)!^2)).
%F A061548 a(n) = 2*A001448(n)/ A117973(n). (End)
%F A061548 a(n) = A001448(n)/A001316(n). - _Peter Luschny_, Mar 23 2014
%F A061548 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
%e A061548 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.
%p A061548 seq(numer(binomial(2*n-1/2, -1/2)), n=0..20);
%t A061548 Table[Numerator[(4*n) !/(2^(4*n)*(2*n) !^2) ], {n, 0, 20}] (* _Indranil Ghosh_, Mar 11 2017 *)
%t A061548 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 *)
%o A061548 (Sage)
%o A061548 def A061548(n): return binomial(4*n,2*n)/2^sum(n.digits(2))
%o A061548 [A061548(n) for n in (0..20)]  # _Peter Luschny_, Mar 23 2014
%o A061548 (PARI) for(n=0, 20, print1(numerator((4*n)!/(2^(4*n)*(2*n)!^2)),", ")) \\ _Indranil Ghosh_, Mar 11 2017
%o A061548 (Python)
%o A061548 import math
%o A061548 f = math.factorial
%o A061548 def A061548(n): return f(4*n) // math.gcd(f(4*n), (2**(4*n)*f(2*n)**2)) # _Indranil Ghosh_, Mar 11 2017
%o A061548 (Magma)
%o A061548 A061548:= func< n | Numerator(Binomial(4*n,2*n)/4^n) >;
%o A061548 [A061548(n): n in [0..25]]; // _G. C. Greubel_, Oct 19 2024
%Y A061548 Cf. A001316, A001448, A061549, A117973.
%Y A061548 Bisection of A001790.
%K A061548 nonn,frac,easy
%O A061548 0,2
%A A061548 Leah Schmelzer (leah2002(AT)mit.edu), May 16 2001
%E A061548 More terms from _Asher Auel_, May 20 2001