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%I #59 Feb 28 2021 03:20:47
%S 2,4,6,10,12,20,24,34,44,64,78,116,148,208,286,410,556,808,1120,1620,
%T 2308,3352,4784,6980,10064,14680,21296,31128,45276,66288,96712,141654,
%U 207156,303716,444748,652612,956884,1404920,2062080,3029564,4450120
%N a(n) = the definite integral Integral_{0..1} Product_{j=1..n} 4*sin^2(Pi*j*x) dx.
%C This quantity arises in some examples associated to the dynamical Mertens's theorem for quasihyperbolic toral automorphisms.
%C The function being integrated to compute a_n vanishes on the set of points in the Farey sequence of level n. I am particularly interested in knowing how large the sequence is asymptotically.
%C a(n) = coefficient of x^(n*(n+1)/2) in the polynomial (-1)^n*Product_{k=1..n} (1-x^k)^2, and is the maximal such coefficient as well. - _Steven Finch_, Feb 03 2009
%H Vaclav Kotesovec, <a href="/A133871/b133871.txt">Table of n, a(n) for n = 1..1000</a> (terms 1..174 from Robert Israel)
%H Miklos Bóna, R. Gómez, M. D. Ward, <a href="http://www.birs.ca/workshops/2016/16w5048/report16w5048.pdf">Workshop in Analytic and Probabilistic Combinatorics BIRS-16w5048</a> 2016.
%H S. R. Finch, <a href="https://www.semanticscholar.org/paper/Signum-Equations-and-Extremal-Coefficients-Finch/66da606fec632a2e752a6f421641fb84148b810f#paperDetail">Signum equations and extremal coefficients</a>.
%H Steven R. Finch, <a href="/A000980/a000980.pdf">Signum equations and extremal coefficients</a>, February 7, 2009. [Cached copy, with permission of the author]
%H Jeffrey Gaither, Guy Louchard, Stephan Wagner, and Mark Daniel Ward, <a href="http://www.stat.purdue.edu/~mdw/papers/paper028.pdf">Resolution of T. Ward's Question and the Israel-Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics</a>, Combinatorics, Probability and Computing, 24 (2015), 195-215. Special Issue Honouring the Memory of Philippe Flajolet.
%H S. Jaidee, S. Stevens and T. Ward, <a href="http://arxiv.org/abs/0801.2082">Mertens' theorem for toral automorphisms</a>, arXiv:0801.2082 [math.DS], 2008-2010.
%H S. Jaidee, S. Stevens and T. Ward, <a href="https://doi.org/10.1090/S0002-9939-2010-10632-9">Mertens' theorem for toral automorphisms</a>, Proc. Amer. Math. Soc. 139 (2011), no. 5, 1819-1824.
%H Mark Daniel Ward, <a href="http://aofa2013.lsi.upc.edu/slides/Ward.pdf">Resolution of T. Ward's Question and the Israel-Finch Conjecture. Precise Asymptotic Analysis of an Integer Sequence Motivated by the Dynamical Mertens' Theorem for Quasihyperbolic Toral Automorphisms</a>, Slides, 2013.
%H T. Ward, D. W. Cantrell and R. Israel, <a href="http://mathforum.org/kb/message.jspa?messageID=6062821">sci.math.research discussion</a>, 2008.
%F a(n) = sum of squares of coefficients in Product_{k=1..n} (1-x^k). - _Paul D. Hanna_, Nov 30 2010
%F a(n) ~ c * d^n / sqrt(n), where d = 1.48770584269062356180051131... and c = 2.40574583936181024... [Ward, 2013]. - _Vaclav Kotesovec_, May 03 2018
%e a(2) = 4 since Integral_{0..1} sin^2(Pi*x) sin^2(2*Pi*x) dx = 1/4.
%p a:= n->int(product(4*(sin(Pi*j*x))^2, j=1..n), x=0..1); seq(a(n), n=1..10);
%p # second Maple program:
%p A133871:= k -> (-1)^k*coeff(mul((t^j-1)^2,j=1..k),t,k*(k+1)/2);
%p # _Robert Israel_, Mar 15 2013
%t p = 1; Table[p = Expand[p*(1 - x^n)^2]; Max[(-1)^n*CoefficientList[p, x]], {n, 1, 100}] (* _Vaclav Kotesovec_, May 03 2018 *)
%t (* The constant "d" *) Chop[-E^(-I*(Pi^2*(1 + 6*x^2) - 6*PolyLog[2, E^(2*I*Pi*x)]) / (6*Pi*x)) /. x -> (x /. FindRoot[Pi*(Pi*(-1 + 6*x^2) + 12*I*x*Log[1 - E^(2*I*Pi*x)]) + 6*PolyLog[2, E^(2*I*Pi*x)], {x, 4/5}, WorkingPrecision -> 100])] (* _Vaclav Kotesovec_, May 04 2018 *)
%o (PARI) a(n)=sum(k=0, n*(n+1)/2, polcoeff(prod(m=1, n, 1-x^m+x*O(x^k)), k)^2) \\ _Paul D. Hanna_
%Y Cf. A005728, A047653.
%K nonn
%O 1,1
%A _Thomas Ward_, Jan 07 2008
%E More terms from _Steven Finch_, Feb 03 2009
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