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A301279 Denominator of variance of n-th row of Pascal's triangle. 4
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 (list; graph; refs; listen; history; text; internal format)
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 A319882

Adjacent sequences:  A301276 A301277 A301278 * A301280 A301281 A301282

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane, Mar 18 2018

STATUS

approved

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Last modified September 19 21:42 EDT 2020. Contains 337184 sequences. (Running on oeis4.)