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

0, 0, 0, 1, 5, 16, 56, 198, 699, 2490, 8943, 32355, 117800, 431316, 1587207, 5867244, 21777203, 81127591, 303240041, 1136914129, 4274441613, 16111746161, 60873695892, 230495640009, 874525192278, 3324270554675, 12658405644200, 48280298159610

Offset: 0

Author

N. J. A. Sloane, Mar 18 2018

Keywords

• nonn

Examples

			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, ...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..1667
• Simon Demers, Taylor's Law Holds for Finite OEIS Integer Sequences and Binomial Coefficients, American Statistician, Volume 72, 2018 - Issue 4.

Crossrefs

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

Programs

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 and A301280
# Alternative:
f:= n -> round((binomial(2*n,n)-4^n/(n+1))/n): f(0):=0:
map(f, [$0..60]); # Robert Israel, Jul 18 2019

Formula

From Robert Israel, Jul 18 2019: (Start)

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

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