login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Numerator of Integral_{x=0..1} Product_{k=1..n} (1-x^k) dx.
5

%I #11 May 24 2015 08:47:13

%S 1,5,41,188,20777,126661,375407075,4551271607,2186878968457691,

%T 405572061653677013,579868609560670025014303,

%U 756499881167742750802544581,90137667815984749912207449629,12095883009361301429642260272492831583,83142433646555338064479023776802561123293

%N Numerator of Integral_{x=0..1} Product_{k=1..n} (1-x^k) dx.

%C Limit n->infinity a(n) / A258230(n) = limit n->infinity Integral_{x=0..1} Product_{k=1..n} (1-x^k) dx = 8*sqrt(3/23)*Pi*sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3)-1) = A258232 = 0.368412535931433652321316597327851...

%H Vaclav Kotesovec, <a href="/A258229/b258229.txt">Table of n, a(n) for n = 1..69</a>

%H StackExchange - Mathematica, <a href="http://mathematica.stackexchange.com/questions/38919/no-response-to-an-infinite-limit">No response to an infinite limit</a>

%e Product_{k=1..n} (1-x^k)

%e n=1 1 - x

%e n=2 1 - x - x^2 + x^3

%e n=3 1 - x - x^2 + x^4 + x^5 - x^6

%e Integral Product_{k=1..n} (1-x^k) dx

%e n=1 x - x^2/2

%e n=2 x - x^2/2 - x^3/3 + x^4/4

%e n=3 x - x^2/2 - x^3/3 + x^5/5 + x^6/6 - x^7/7

%e For Integral_{x=0..1} set x=1

%e n=1 1 - 1/2 = 1/2, a(1) = 1

%e n=2 1 - 1/2 - 1/3 + 1/4 = 5/12, a(2) = 5

%e n=3 1 - 1/2 - 1/3 + 1/5 + 1/6 - 1/7 = 41/105, a(3) = 41

%t nmax=15; p=1; Table[p=Expand[p*(1-x^n)]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]], {n,1,nmax}] // Numerator

%Y Cf. A258230, A258191, A258192, A258232.

%K nonn

%O 1,2

%A _Vaclav Kotesovec_, May 24 2015