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

0,3

Comments

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

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.

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.

Example

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, ...

Prog

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

Crossrefs

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

Sequence in context: A095835 A147805 A141527 * A191298 A104025 A335600

Adjacent sequences: A301628 A301629 A301630 * A301632 A301633 A301634

Keyword

nonn,frac

Author

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

Status

approved