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

0,4

Comments

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

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.

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

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 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
(PARI) a(n) = if(n==0, 0, numerator(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: A065777 A193485 A006422 * A186677 A277654 A247767

Adjacent sequences: A301275 A301276 A301277 * A301279 A301280 A301281

Keyword

nonn,frac

Author

N. J. A. Sloane, Mar 18 2018

Status

approved