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A301280 Nearest integer to variance of n-th row of Pascal's triangle. 4

%I

%S 0,0,0,1,5,16,56,198,699,2490,8943,32355,117800,431316,1587207,

%T 5867244,21777203,81127591,303240041,1136914129,4274441613,

%U 16111746161,60873695892,230495640009,874525192278,3324270554675,12658405644200,48280298159610

%N Nearest integer to variance of n-th row of Pascal's triangle.

%D Simon Demers, Taylor's Law Holds for Finite OEIS Integer Sequences and Binomial Coefficients, American Statistician, online: 19 Jan 2018; https://doi.org/10.1080/00031305.2017.1422439

%H Robert Israel, <a href="/A301280/b301280.txt">Table of n, a(n) for n = 0..1667</a>

%F From _Robert Israel_, Jul 18 2019: (Start)

%F The variance is binomial(2*n,n)/n - 4^n/(n*(n+1)).

%F a(n) ~ 4^n/((sqrt(Pi)*n^(3/2)). (End)

%e 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, ...

%p M:=70;

%p m := n -> 2^n/(n+1);

%p m1:=[seq(m(n),n=0..M)]; # A084623/A000265

%p v := n -> (1/n) * add((binomial(n,i) - m(n))^2, i=0..n );

%p v1:= [0, 0, seq(v(n),n=2..60)]; # A301278/A301279 and A301280

%p # Alternative:

%p f:= n -> round((binomial(2*n,n)-4^n/(n+1))/n): f(0):=0:

%p map(f, [$0..60]); # _Robert Israel_, Jul 18 2019

%Y Mean and variance of n-th row of Pascal's triangle: A084623/A000265, A301278/A301279, A054650.

%K nonn

%O 0,5

%A _N. J. A. Sloane_, Mar 18 2018

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Last modified May 12 21:43 EDT 2021. Contains 343829 sequences. (Running on oeis4.)