A301631 Numerator of population variance of n-th row of Pascal's triangle.

0, 0, 2, 1, 94, 122, 2372, 173, 50294, 56014, 983740, 266930, 18376812, 19624884, 333313544, 5500541, 5923399334, 6206260694, 103708093964, 27001710566, 1795265477444, 1860906681644, 30802090121144, 1988024895074, 524715115366844, 540193965134732, 8886200762228312

Offset: 0

Author

N. J. A. Sloane and Chai Wah Wu, Mar 24 2018

Keywords

• nonn
• frac

Comments

Denominator of population variance of n-th row of Pascal's triangle is A191871(n+1) = A000265(n+1)^2.

Examples

			The first few population variances are 0, 0, 2/9, 1, 94/25, 122/9, 2372/49, 173, 50294/81, 56014/25, 983740/121, 266930/9, 18376812/169, 19624884/49, 333313544/225, 5500541, 5923399334/289, ...
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..1659
• Simon Demers, Taylor's Law Holds for Finite OEIS Integer Sequences and Binomial Coefficients, American Statistician, online: 19 Jan 2018.

Crossrefs

Cf. A000108, A075101, A084623, A000265, A191871 (denominators), A301278, A301279, A054650, A301280.

Programs

PARI

a(n) = numerator(binomial(2*n,n)/(n+1) - 4^n/(n+1)^2); \\ Altug Alkan, Mar 25 2018

Python

from fractions import Fraction
from sympy import binomial
def A301631(n):
    return (Fraction(int(binomial(2*n,n)))/(n+1) - Fraction(4**n)/(n+1)**2).numerator

Formula

a(n) = numerator of binomial(2n,n)/(n+1) - 4^n/(n+1)^2.

a(n) = A000108(n)*A000265(n+1)^2 - A075101(n+1)^2/4.