login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Numerator of rational part of raw moment n of the line point picking problem.
1

%I #28 Jan 06 2023 03:21:25

%S -1,3,-4,17,-41,42,-289,1171,-1739,1753,-19157,19262,-249251,250241,

%T -249383,200107,-1696405,1700409,-32239703,161504821,-161227687,

%U 161479627,-3708740681,3713590526,-18545643343,18566236531,-55641506293,55694623643,-230529988171

%N Numerator of rational part of raw moment n of the line point picking problem.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LinePointPicking.html">Line Point Picking</a>

%F From _Pontus von Brömssen_, Nov 03 2019: (Start)

%F For even n, a(n)/A115389(n) = 2*n*Sum_{k = n/2..n-1} 1/k - 1.

%F For odd n >= 3, a(n)/A115389(n) = -2*n*((Sum_{k = (n-1)/2..n-2} 1/k) - 1/(n-1)) - 1. (End)

%F a(n) = numerator of 1 + (2*n)*Sum_{k = 1..n} (-1)^(n+k+1)/k. - _Peter Bala_, Jan 05 2023

%F a(n) = numerator of 2*n*((-1)^n*log(2) - LerchPhi(-1, 1, n + 1)) + 1. - _Peter Luschny_, Jan 05 2023

%e -1 + 2*log(2), 3 - 4*log(2), -4 + 6*log(2), 17/3 - 8*log(2), -41/6 + 10*log(2), ...

%e The above sequence of numbers is given by 4*Integral_{x = 0..Pi/4} tan(x)^(2*n+1) * cos(x)^2 dx for n >= 1, or, equivalently, by Integral_{y = 0..1} Integral_{x = 0..1} 2*n*(x*y)^n/(x + y)^2 dx dy for n >= 1. - _Peter Bala_, Jan 04 2023

%p a := n -> numer(1 + 2*n*add((-1)^(n+k+1)/k, k = 1..n)):

%p seq(a(n), n = 1..28); # _Peter Bala_, Jan 05 2023

%p # Alternative:

%p a := n -> 2*n*((-1)^n*log(2) - LerchPhi(-1, 1, n + 1)) + 1:

%p seq(numer(simplify(a(n))), n = 1..29); # _Peter Luschny_, Jan 05 2023

%Y Cf. A058312, A058313, A115389.

%K sign,frac,easy

%O 1,2

%A _Eric W. Weisstein_, Jan 21 2006