A115388 Numerator of rational part of raw moment n of the line point picking problem.

-1, 3, -4, 17, -41, 42, -289, 1171, -1739, 1753, -19157, 19262, -249251, 250241, -249383, 200107, -1696405, 1700409, -32239703, 161504821, -161227687, 161479627, -3708740681, 3713590526, -18545643343, 18566236531, -55641506293, 55694623643, -230529988171

Offset

1,2

Links

Table of n, a(n) for n=1..29.

Eric Weisstein's World of Mathematics, Line Point Picking

Formula

From Pontus von Brömssen, Nov 03 2019: (Start)

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

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

a(n) = numerator of 1 + (2*n)*Sum_{k = 1..n} (-1)^(n+k+1)/k. - Peter Bala, Jan 05 2023

a(n) = numerator of 2*n*((-1)^n*log(2) - LerchPhi(-1, 1, n + 1)) + 1. - Peter Luschny, Jan 05 2023

Example

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

Maple

a := n -> numer(1 + 2*n*add((-1)^(n+k+1)/k, k = 1..n)):
seq(a(n), n = 1..28); # Peter Bala, Jan 05 2023
# Alternative:
a := n -> 2*n*((-1)^n*log(2) - LerchPhi(-1, 1, n + 1)) + 1:
seq(numer(simplify(a(n))), n = 1..29); # Peter Luschny, Jan 05 2023

Crossrefs

Cf. A058312, A058313, A115389.

Sequence in context: A100434 A096876 A257330 * A187995 A296277 A303480

Adjacent sequences: A115385 A115386 A115387 * A115389 A115390 A115391

Keyword

sign,frac,easy

Author

Eric W. Weisstein, Jan 21 2006

Status

approved