cp's OEIS Frontend

A301279 Denominator of variance of n-th row of Pascal's triangle.

%I A301279 #24 Mar 26 2018 04:24:37
%S A301279 1,1,3,3,10,15,21,7,36,45,11,33,13,91,105,15,136,153,171,95,105,231,
%T A301279 253,69,150,325,351,189,203,435,155,31,528,51,595,315,111,703,741,195,
%U A301279 410,861,903,473,495,1035,1081,141,588,1225,255,663,689,1431,1485
%N A301279 Denominator of variance of n-th row of Pascal's triangle.
%C A301279 Variance here is sample variance unbiased estimator. For population variance, the denominator is A191871(n+1) = A000265(n+1)^2. - _Chai Wah Wu_, Mar 25 2018
%H A301279 Chai Wah Wu, <a href="/A301279/b301279.txt">Table of n, a(n) for n = 0..10000</a>
%H A301279 Simon Demers, <a href="https://doi.org/10.1080/00031305.2017.1422439">Taylor's Law Holds for Finite OEIS Integer Sequences and Binomial Coefficients</a>, American Statistician, online: 19 Jan 2018.
%F A301279 a(0) = 1; a(n) = denominator of binomial(2n,n)/n - 4^n/(n*(n+1)) for n >= 1. - _Chai Wah Wu_, Mar 23 2018
%e A301279 The first few variances are 0, 0, 1/3, 4/3, 47/10, 244/15, 1186/21, 1384/7, 25147/36, 112028/45, 98374/11, 1067720/33, 1531401/13, 39249768/91, 166656772/105, 88008656/15, 2961699667/136, 12412521388/153, 51854046982/171, 108006842264/95, 448816369361/105, ...
%p A301279 M:=70;
%p A301279 m := n -> 2^n/(n+1);
%p A301279 m1:=[seq(m(n),n=0..M)]; # A084623/A000265
%p A301279 v := n -> (1/n) * add((binomial(n,i) - m(n))^2, i=0..n );
%p A301279 v1:= [0, 0, seq(v(n),n=2..60)]; # A301278/A301279
%o A301279 (Python)
%o A301279 from fractions import Fraction
%o A301279 from sympy import binomial
%o A301279 def A301279(n):
%o A301279     return (Fraction(int(binomial(2*n,n)))/n - Fraction(4**n)/(n*(n+1))).denominator if n > 0 else 1 # _Chai Wah Wu_, Mar 23 2018
%o A301279 (PARI) a(n) = if(n==0, 1, denominator(binomial(2*n,n)/n - 4^n/(n*(n+1)))); \\ _Altug Alkan_, Mar 25 2018
%Y A301279 Mean and variance of n-th row of Pascal's triangle: A084623/A000265, A301278/A301279, A054650, A301280.
%K A301279 nonn,frac
%O A301279 0,3
%A A301279 _N. J. A. Sloane_, Mar 18 2018