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A301278 Numerator of variance of n-th row of Pascal's triangle. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Variance here is the sample variance unbiased estimator. For population variance, see A301631.
LINKS
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
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Mar 18 2018
STATUS
approved

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Last modified May 18 19:36 EDT 2024. Contains 372666 sequences. (Running on oeis4.)