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A301278
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Numerator of variance of n-th row of Pascal's triangle.
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4
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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
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OFFSET
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0,4
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COMMENTS
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Variance here is the sample variance unbiased estimator. For population variance, see A301631.
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LINKS
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FORMULA
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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
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EXAMPLE
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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, ...
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MAPLE
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M:=70;
m := n -> 2^n/(n+1);
v := n -> (1/n) * add((binomial(n, i) - m(n))^2, i=0..n );
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PROG
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(Python)
from fractions import Fraction
from sympy import binomial
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
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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