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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 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

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