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

1, 1, 3, 3, 10, 15, 21, 7, 36, 45, 11, 33, 13, 91, 105, 15, 136, 153, 171, 95, 105, 231, 253, 69, 150, 325, 351, 189, 203, 435, 155, 31, 528, 51, 595, 315, 111, 703, 741, 195, 410, 861, 903, 473, 495, 1035, 1081, 141, 588, 1225, 255, 663, 689, 1431, 1485

Offset: 0

Author

N. J. A. Sloane, Mar 18 2018

Keywords

• nonn
• frac

Comments

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

Examples

			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, ...
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000
• Simon Demers, Taylor's Law Holds for Finite OEIS Integer Sequences and Binomial Coefficients, American Statistician, online: 19 Jan 2018.

Crossrefs

Mean and variance of n-th row of Pascal's triangle: A084623/A000265, A301278/A301279, A054650, A301280.

Programs

Maple

M:=70;
m := n -> 2^n/(n+1);
m1:=[seq(m(n),n=0..M)]; # A084623/A000265
v := n -> (1/n) * add((binomial(n,i) - m(n))^2, i=0..n );
v1:= [0, 0, seq(v(n),n=2..60)]; # A301278/A301279

PARI

a(n) = if(n==0, 1, denominator(binomial(2*n,n)/n - 4^n/(n*(n+1)))); \\ Altug Alkan, Mar 25 2018

Python

from fractions import Fraction
from sympy import binomial
def A301279(n):
    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

Formula

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