cp's OEIS Frontend

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

%I A301280 #22 Nov 25 2024 05:01:18
%S A301280 0,0,0,1,5,16,56,198,699,2490,8943,32355,117800,431316,1587207,
%T A301280 5867244,21777203,81127591,303240041,1136914129,4274441613,
%U A301280 16111746161,60873695892,230495640009,874525192278,3324270554675,12658405644200,48280298159610
%N A301280 Nearest integer to variance of n-th row of Pascal's triangle.
%H A301280 Robert Israel, <a href="/A301280/b301280.txt">Table of n, a(n) for n = 0..1667</a>
%H A301280 Simon Demers, <a href="https://doi.org/10.1080/00031305.2017.1422439">Taylor's Law Holds for Finite OEIS Integer Sequences and Binomial Coefficients</a>, American Statistician, Volume 72, 2018 - Issue 4.
%F A301280 From _Robert Israel_, Jul 18 2019: (Start)
%F A301280 The variance is binomial(2*n,n)/n - 4^n/(n*(n+1)).
%F A301280 a(n) ~ 4^n/(sqrt(Pi)*n^(3/2)). (End)
%e A301280 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 A301280 M:=70;
%p A301280 m := n -> 2^n/(n+1);
%p A301280 m1:=[seq(m(n),n=0..M)]; # A084623/A000265
%p A301280 v := n -> (1/n) * add((binomial(n,i) - m(n))^2, i=0..n );
%p A301280 v1:= [0, 0, seq(v(n),n=2..60)]; # A301278/A301279 and A301280
%p A301280 # Alternative:
%p A301280 f:= n -> round((binomial(2*n,n)-4^n/(n+1))/n): f(0):=0:
%p A301280 map(f, [$0..60]); # _Robert Israel_, Jul 18 2019
%Y A301280 Mean and variance of n-th row of Pascal's triangle: A084623/A000265, A301278/A301279, A054650.
%K A301280 nonn
%O A301280 0,5
%A A301280 _N. J. A. Sloane_, Mar 18 2018