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A301280 Nearest integer to variance of n-th row of Pascal's triangle. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

REFERENCES

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

LINKS

Robert Israel, Table of n, a(n) for n = 0..1667

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, A301280.

Sequence in context: A299685 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

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Last modified September 27 03:08 EDT 2020. Contains 337380 sequences. (Running on oeis4.)