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A301631 Numerator of population variance of n-th row of Pascal's triangle. 2
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 (list; graph; refs; listen; history; text; internal format)
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 A039923

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

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Last modified January 23 13:40 EST 2020. Contains 331171 sequences. (Running on oeis4.)