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

0,3

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

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.

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

Example

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, ...

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

Prog

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

Crossrefs

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

Sequence in context: A129885 A277963 A193965 * A330288 A262923 A367301

Adjacent sequences: A301276 A301277 A301278 * A301280 A301281 A301282

Keyword

nonn,frac

Author

N. J. A. Sloane, Mar 18 2018

Status

approved