A258230 Denominator of Integral_{x=0..1} Product_{k=1..n} (1-x^k) dx.

2, 12, 105, 495, 55440, 340340, 1012647636, 12304749600, 5920545668637600, 1098951951860282520, 1572101004939647757775200, 2051717579526635495717258016, 244523633377266327241371614400, 32818916025992059215981780272862841200

Offset: 1

Author

Vaclav Kotesovec, May 24 2015

Keywords

• nonn

Comments

Limit n->infinity A258229(n) / a(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...

Examples

			Product_{k=1..n} (1-x^k)
n=1 1 - x
n=2 1 - x - x^2 + x^3
n=3 1 - x - x^2 + x^4 + x^5 - x^6
Integral Product_{k=1..n} (1-x^k) dx
n=1 x - x^2/2
n=2 x - x^2/2 - x^3/3 + x^4/4
n=3 x - x^2/2 - x^3/3 + x^5/5 + x^6/6 - x^7/7
For Integral_{x=0..1} set x=1
n=1 1 - 1/2 = 1/2, a(1) = 2
n=2 1 - 1/2 - 1/3 + 1/4 = 5/12, a(2) = 12
n=3 1 - 1/2 - 1/3 + 1/5 + 1/6 - 1/7 = 41/105, a(3) = 105
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..69
• StackExchange - Mathematica, No response to an infinite limit

Crossrefs

Cf. A258229, A258191, A258192, A258232.

Programs

Mathematica

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