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

0,5

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.

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)

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 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

Crossrefs

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

Sequence in context: A363449 A268225 A120343 * A153366 A057553 A226973

Adjacent sequences: A301277 A301278 A301279 * A301281 A301282 A301283

Keyword

nonn

Author

N. J. A. Sloane, Mar 18 2018

Status

approved