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A133871
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a(n) = the definite integral Integral_{0..1} Product_{j=1..n} 4*sin^2(Pi*j*x) dx.
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3
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2, 4, 6, 10, 12, 20, 24, 34, 44, 64, 78, 116, 148, 208, 286, 410, 556, 808, 1120, 1620, 2308, 3352, 4784, 6980, 10064, 14680, 21296, 31128, 45276, 66288, 96712, 141654, 207156, 303716, 444748, 652612, 956884, 1404920, 2062080, 3029564, 4450120
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OFFSET
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1,1
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COMMENTS
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This quantity arises in some examples associated to the dynamical Mertens's theorem for quasihyperbolic toral automorphisms.
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.
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
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 1..1000 (terms 1..174 from Robert Israel)
Miklos Bóna, R. Gómez, M. D. Ward, Workshop in Analytic and Probabilistic Combinatorics BIRS-16w5048 2016.
S. R. Finch, Signum equations and extremal coefficients.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
Jeffrey Gaither, Guy Louchard, Stephan Wagner, and Mark Daniel Ward, Resolution of T. Ward's Question and the Israel-Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics, Combinatorics, Probability and Computing, 24 (2015), 195-215. Special Issue Honouring the Memory of Philippe Flajolet.
S. Jaidee, S. Stevens and T. Ward, Mertens' theorem for toral automorphisms, arXiv:0801.2082 [math.DS], 2008-2010. - Steven Finch, Feb 06 2009
Mark Daniel Ward, 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, Slides, 2013.
T. Ward, D. W. Cantrell and R. Israel, sci.math.research discussion, 2008.
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FORMULA
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a(n) = sum of squares of coefficients in Product_{k=1..n} (1-x^k). - Paul D. Hanna, Nov 30 2010
a(n) ~ c * d^n / sqrt(n), where d = 1.48770584269062356180051131... and c = 2.40574583936181024... [Ward, 2013]. - Vaclav Kotesovec, May 03 2018
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EXAMPLE
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a(2) = 4 since Integral_{0..1} sin^2(Pi*x) sin^2(2*Pi*x) dx = 1/4.
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MAPLE
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a:= n->int(product(4*(sin(Pi*j*x))^2, j=1..n), x=0..1); seq(a(n), n=1..10);
# second Maple program:
A133871:= k -> (-1)^k*coeff(mul((t^j-1)^2, j=1..k), t, k*(k+1)/2);
# Robert Israel, Mar 15 2013
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MATHEMATICA
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p = 1; Table[p = Expand[p*(1 - x^n)^2]; Max[(-1)^n*CoefficientList[p, x]], {n, 1, 100}] (* Vaclav Kotesovec, May 03 2018 *)
(* 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 *)
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PROG
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(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
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CROSSREFS
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Cf. A005728, A047653.
Sequence in context: A068336 A293821 A194944 * A068514 A074645 A125286
Adjacent sequences: A133868 A133869 A133870 * A133872 A133873 A133874
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KEYWORD
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nonn
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AUTHOR
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Thomas Ward (t.ward(AT)uea.ac.uk), Jan 07 2008
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EXTENSIONS
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More terms from Steven Finch, Feb 03 2009
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STATUS
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approved
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