A301278 Numerator of variance of n-th row of Pascal's triangle.

0, 0, 1, 4, 47, 244, 1186, 1384, 25147, 112028, 98374, 1067720, 1531401, 39249768, 166656772, 88008656, 2961699667, 12412521388, 51854046982, 108006842264, 448816369361, 3721813363288, 15401045060572, 15904199160592, 131178778841711, 1080387930269464, 4443100381114156, 9124976352166288

Offset: 0

Author

N. J. A. Sloane, Mar 18 2018

Keywords

• nonn
• frac

Comments

Variance here is the sample variance unbiased estimator. For population variance, see A301631.

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..1659
• 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, 0, numerator(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 A301278(n):
    return (Fraction(int(binomial(2*n,n)))/n - Fraction(4**n)/(n*(n+1))).numerator if n > 0 else 0 # Chai Wah Wu, Mar 23 2018

Formula

a(0) = 0; a(n) = numerator of binomial(2n,n)/n - 4^n/(n*(n+1)) for n >= 1. - Chai Wah Wu, Mar 23 2018