A177387 E.g.f.: Sum_{n>=0} Product_{k=1..n} sin(k*x).
1, 1, 4, 35, 536, 12721, 432364, 19923455, 1195597616, 90597432961, 8459910749524, 954441965659775, 127987398340965896, 20120987017230590401, 3665273670382984503484, 765857737574513717138495
Offset: 0
Examples
E.g.f: A(x) = 1 + x + 4*x^2/2! + 35*x^3/3! + 536*x^4/4! +... A(x) = 1 + sin(x) + sin(x)*sin(2x) + sin(x)*sin(2x)*sin(3x) + ...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..253
- Vaclav Kotesovec, Graph of 1000 terms (limit to constant c)
Programs
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Mathematica
Flatten[{1,Rest[CoefficientList[Series[Sum[Product[Sin[m*x],{m,1,k}],{k,1,20}],{x,0,20}],x] * Range[0,20]!]}] (* Vaclav Kotesovec, Nov 03 2014 *) nn=20; tab=ConstantArray[0,nn]; tab[[1]]=Series[Sin[x],{x,0,nn}]; Do[tab[[k]]=Series[tab[[k-1]]*Sin[k*x],{x,0,nn}],{k,2,nn}]; Flatten[{1,Rest[CoefficientList[Sum[tab[[k]],{k,1,nn}],x]*Range[0,nn]!]}] (* Vaclav Kotesovec, Nov 03 2014 (more efficient) *) -
PARI
{a(n)=local(X=x+x*O(x^n),Egf);Egf=sum(m=0,n,prod(k=1,m,sin(k*X)));n!*polcoeff(Egf,n)}
Formula
a(n) ~ c * 2^(n+1) * n^(2*n+7/6) / (Pi^(n-1) * exp(2*n) * (log(2))^n), where c = 1.01529686... . - Vaclav Kotesovec, Nov 03 2014