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%I #16 Mar 31 2018 14:41:48
%S 0,0,2,1,94,122,2372,173,50294,56014,983740,266930,18376812,19624884,
%T 333313544,5500541,5923399334,6206260694,103708093964,27001710566,
%U 1795265477444,1860906681644,30802090121144,1988024895074,524715115366844,540193965134732,8886200762228312
%N Numerator of population variance of n-th row of Pascal's triangle.
%C Denominator of population variance of n-th row of Pascal's triangle is A191871(n+1) = A000265(n+1)^2.
%H Chai Wah Wu, <a href="/A301631/b301631.txt">Table of n, a(n) for n = 0..1659</a>
%H Simon Demers, <a href="https://doi.org/10.1080/00031305.2017.1422439">Taylor's Law Holds for Finite OEIS Integer Sequences and Binomial Coefficients</a>, American Statistician, online: 19 Jan 2018.
%F a(n) = numerator of binomial(2n,n)/(n+1) - 4^n/(n+1)^2.
%F a(n) = A000108(n)*A000265(n+1)^2 - A075101(n+1)^2/4.
%e 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, ...
%o (Python)
%o from fractions import Fraction
%o from sympy import binomial
%o def A301631(n):
%o return (Fraction(int(binomial(2*n,n)))/(n+1) - Fraction(4**n)/(n+1)**2).numerator
%o (PARI) a(n) = numerator(binomial(2*n,n)/(n+1) - 4^n/(n+1)^2); \\ _Altug Alkan_, Mar 25 2018
%Y Cf. A000108, A075101, A084623, A000265, A191871 (denominators), A301278, A301279, A054650, A301280.
%K nonn,frac
%O 0,3
%A _N. J. A. Sloane_ and _Chai Wah Wu_, Mar 24 2018
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